{"id":1069,"date":"2026-01-13T13:15:17","date_gmt":"2026-01-13T03:15:17","guid":{"rendered":"https:\/\/archive4ones.com\/2ndstudy\/?p=1069"},"modified":"2026-01-13T13:15:17","modified_gmt":"2026-01-13T03:15:17","slug":"year11-math-2-1-4-applied-trigonometry","status":"publish","type":"post","link":"https:\/\/archive4ones.com\/2ndstudy\/?p=1069","title":{"rendered":"Year11 MATH 2-1-4 Applied Trigonometry"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Applied Trigonometry in Year 11 Maths extends basic right-angled triangle ratios (sin cos tan) to more complex areas like <strong>radians, unit circle, trigonometric graphs, identities, and the Sine\/Cosine Rules<\/strong>, allowing you to solve problems in non-right-angled triangles, find areas, arc lengths, and model periodic behaviour, moving beyond just finding sides and angles in simple triangles to real-world applications.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Here&#8217;s a breakdown of key concepts:&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Beyond Right-Angled Triangles:<\/strong> You learn to use the <strong>Sine Rule<\/strong> (for any triangle) and the <strong>Cosine Rule<\/strong> (for any triangle) to find unknown sides and angles.<\/li>\n\n\n\n<li><strong>Radians:<\/strong> Understand angles in radians, not just degrees, and how to convert between them, crucial for calculus and advanced functions.<\/li>\n\n\n\n<li><strong>Unit Circle &amp; General Angles:<\/strong> Extend sin, cos, tan to any angle (positive, negative, over 360\u00b0) using the unit circle, understanding ASTC (All Students Take Calculus).<\/li>\n\n\n\n<li><strong>Trigonometric Graphs:<\/strong> Graph y = sin(x), y = cos(x), y = tan(x), exploring transformations (amplitude, period, phase shift) to model waves, sound, and oscillations.<\/li>\n\n\n\n<li><strong>Trigonometric Identities:<\/strong> Prove and use identities like <math data-latex=\"\\sin ^{2}\\theta +\\cos ^{2}\\theta =1\"><semantics><mrow><msup><mi>sin<\/mi><mn>2<\/mn><\/msup><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><mi>\u03b8<\/mi><mo>+<\/mo><msup><mi>cos<\/mi><mn>2<\/mn><\/msup><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><mi>\u03b8<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\sin ^{2}\\theta +\\cos ^{2}\\theta =1<\/annotation><\/semantics><\/math>    to simplify complex expressions.<\/li>\n\n\n\n<li><strong>Applications:<\/strong> Solve problems involving sectors, arc lengths, bearings, navigation, and more complex geometric scenarios.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Essentially, Year 11 Applied Trigonometry builds a powerful toolkit to describe periodic phenomena and solve complex measurement problems in geometry and physics.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">*****************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>The Sine Rule&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The <strong>Sine Rule<\/strong> (or Law of Sines) is used in any triangle to find unknown sides when you have corresponding angles, or to find unknown angles when you have corresponding sides and another angle. It establishes a relationship between the sides of a non-right-angled triangle and the sines of their opposite angles.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The formula is expressed as: &nbsp;<math data-latex=\"\\frac{a}{\\sin A}=\\frac{b}{\\sin B}=\\frac{c}{\\sin C}\"><semantics><mrow><mfrac><mi>a<\/mi><mrow><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>A<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mi>b<\/mi><mrow><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mi>c<\/mi><mrow><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>C<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{a}{\\sin A}=\\frac{b}{\\sin B}=\\frac{c}{\\sin C}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Where <math data-latex=\"a,b\"><semantics><mrow><mi>a<\/mi><mo separator=\"true\">,<\/mo><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a,b<\/annotation><\/semantics><\/math> and <math data-latex=\"c\"><semantics><mi>c<\/mi><annotation encoding=\"application\/x-tex\">c<\/annotation><\/semantics><\/math> are the lengths of the sides opposite to angles <math data-latex=\"A,B\"><semantics><mrow><mi>A<\/mi><mo separator=\"true\">,<\/mo><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A,B<\/annotation><\/semantics><\/math> and <math data-latex=\"C\"><semantics><mi>C<\/mi><annotation encoding=\"application\/x-tex\">C<\/annotation><\/semantics><\/math>, respectively.&nbsp;<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>The Cosine Rule&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The <strong>Cosine Rule<\/strong> (or Law of Cosines) generalizes the Pythagorean theorem to any triangle. It is primarily used to find the third side of a triangle when two sides and the included angle are known (SAS), or to find the angles of a triangle when all three sides are known (SSS).&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The formula can be expressed in three forms, depending on which side you want to find:&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math data-latex=\"a^{2}=b^{2}+c^{2}-2bc\\cos A\"><semantics><mrow><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>c<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>2<\/mn><mi>b<\/mi><mi>c<\/mi><mrow><mspace width=\"0.1667em\"><\/mspace><mi>cos<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a^{2}=b^{2}+c^{2}-2bc\\cos A<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math data-latex=\"b^{2}=a^{2}+c^{2}-2ac\\cos B\"><semantics><mrow><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>c<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>2<\/mn><mi>a<\/mi><mi>c<\/mi><mrow><mspace width=\"0.1667em\"><\/mspace><mi>cos<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">b^{2}=a^{2}+c^{2}-2ac\\cos B<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math data-latex=\"c^{2}=a^{2}+b^{2}-2ab\\cos C\"><semantics><mrow><msup><mi>c<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>2<\/mn><mi>a<\/mi><mi>b<\/mi><mrow><mspace width=\"0.1667em\"><\/mspace><mi>cos<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">c^{2}=a^{2}+b^{2}-2ab\\cos C<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">It can also be rearranged to solve for an angle, for example, angle <math data-latex=\"A\"><semantics><mi>A<\/mi><annotation encoding=\"application\/x-tex\">A<\/annotation><\/semantics><\/math>:<math data-latex=\"\\cos A=\\frac{b^{2}+c^{2}-a^{2}}{2bc}\"><semantics><mrow><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>A<\/mi><mo>=<\/mo><mfrac><mrow><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>c<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><msup><mi>a<\/mi><mn>2<\/mn><\/msup><\/mrow><mrow><mn>2<\/mn><mi>b<\/mi><mi>c<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\cos A=\\frac{b^{2}+c^{2}-a^{2}}{2bc}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">********************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>radians<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In QLD Year 11 Maths (Mathematical Methods), radians are a standard way to measure angles, defined as the angle where the arc length equals the radius; a full circle is radians (360\u00b0), making \u03c0 radians equal to 180\u00b0. Radians are crucial for calculus (differentiation of trig functions) and use \u03c0 (approx. 3.14) instead of degrees, with conversions like multiplying degrees by <math data-latex=\"\\frac{\\pi }{180}\"><semantics><mfrac><mi>\u03c0<\/mi><mn>180<\/mn><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac{\\pi }{180}<\/annotation><\/semantics><\/math> to get radians, and are used in the unit circle for circular functions.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Key Concepts&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Definition<\/strong>: One radian is the angle subtended at the center of a circle when the arc length is exactly the same as the radius.<\/li>\n\n\n\n<li><strong>Relationship to <\/strong><strong>\u03c0<\/strong>:\n<ul class=\"wp-block-list\">\n<li>2\u03c0 radians = 360\u00b0 (a full circle)<\/li>\n\n\n\n<li>\u03c0 radians = 180\u00b0 (a straight line\/semicircle)<\/li>\n\n\n\n<li><math data-latex=\"\\frac{\\pi }{2}\"><semantics><mfrac><mi>\u03c0<\/mi><mn>2<\/mn><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac{\\pi }{2}<\/annotation><\/semantics><\/math> radians = 90\u00b0 (a right angle)<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Why Radians?<\/strong>: They simplify formulas and are essential for higher-level mathematics, especially when dealing with calculus and circular functions (trigonometry).<\/li>\n\n\n\n<li><strong>Unit<\/strong>: Radians don&#8217;t usually have a symbol, but sometimes a small &#8216;c&#8217; (for circular) or &#8216;rad&#8217; is used, though often no symbol implies radians in mathematical contexts.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Conversion Formulas&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Degrees to Radians<\/strong>: Multiply the angle (in degrees) by <math data-latex=\"\\frac{\\pi }{180}\"><semantics><mfrac><mi>\u03c0<\/mi><mn>180<\/mn><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac{\\pi }{180}<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><strong>Radians to Degrees<\/strong>: Multiply the angle (in radians) by <math data-latex=\"\\frac {180}{\\pi }\"><semantics><mfrac><mn>180<\/mn><mi>\u03c0<\/mi><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac {180}{\\pi }<\/annotation><\/semantics><\/math>.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">In Practice&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Calculator Mode<\/strong>: Ensure your calculator is in &#8216;RAD&#8217; mode when working with radians.<\/li>\n\n\n\n<li><strong>Unit Circle<\/strong>: Angles on the unit circle (radius 1) are easily expressed in radians (x-axis values become \u03c0, 2\u03c0 , etc.).&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">A circle with a radius of 1 is called a <strong>unit circle<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Usually, a unit circle is drawn with its center at the origin on a coordinate plane (see the diagram on the right). If the angle between the axis and the line segment is as shown in the diagram, the coordinates of a point on the unit circle are as follows, according to the definition of trigonometric functions:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"278\" height=\"279\" src=\"https:\/\/archive4ones.com\/2ndstudy\/wp-content\/uploads\/sites\/8\/2026\/01\/Unit-Circle.png\" alt=\"\" class=\"wp-image-1071\" srcset=\"https:\/\/archive4ones.com\/2ndstudy\/wp-content\/uploads\/sites\/8\/2026\/01\/Unit-Circle.png 278w, https:\/\/archive4ones.com\/2ndstudy\/wp-content\/uploads\/sites\/8\/2026\/01\/Unit-Circle-150x150.png 150w\" sizes=\"auto, (max-width: 278px) 100vw, 278px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">****************************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Trigonomrtric Graphs<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In Queensland Year 11 Mathematics, trigonometric graphs are&nbsp;&nbsp;<strong>Mathematical Methods<\/strong>&nbsp;applied to functions and relations. They are used to model periodic or cyclic real-world phenomena like ocean tides or pendulum swings. In&nbsp;<strong>General Mathematics<\/strong>, the focus is more on applications of trigonometry in right-angled and non-right-angled triangles rather than sketching and transforming the graphs themselves.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>&gt;&gt; For Mathematical Methods Students&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The Year 11 Mathematical Methods syllabus covers a detailed study of trigonometric functions and their corresponding graphs, including:&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Understanding the Unit Circle<\/strong>: Defining the sine (sin <math data-latex=\"\\theta\"><semantics><mi>\u03b8<\/mi><annotation encoding=\"application\/x-tex\">\\theta<\/annotation><\/semantics><\/math>), cosine (cos <math data-latex=\"\\theta\"><semantics><mi>\u03b8<\/mi><annotation encoding=\"application\/x-tex\">\\theta<\/annotation><\/semantics><\/math>), and tangent (tan <math data-latex=\"\\theta\"><semantics><mi>\u03b8<\/mi><annotation encoding=\"application\/x-tex\">\\theta<\/annotation><\/semantics><\/math>) functions using the unit circle and understanding their periodic nature.<\/li>\n\n\n\n<li><strong>Angle Measurement<\/strong>: Using both degrees and radians, and converting between them (e.g., 360\u00b0 = 2\u03c0 radians).<\/li>\n\n\n\n<li><strong>Sketching Basic Graphs<\/strong>: Recognising and sketching the graphs of <math data-latex=\"y = sin(x)\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>s<\/mi><mi>i<\/mi><mi>n<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">y = sin(x)<\/annotation><\/semantics><\/math>, <math data-latex=\"y = cos(x)\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>c<\/mi><mi>o<\/mi><mi>s<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">y = cos(x)<\/annotation><\/semantics><\/math>, and <math data-latex=\"y = tan(x)\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>t<\/mi><mi>a<\/mi><mi>n<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">y = tan(x)<\/annotation><\/semantics><\/math>over extended domains.<\/li>\n\n\n\n<li><strong>Key Features<\/strong>: Identifying the properties of these graphs:\n<ul class=\"wp-block-list\">\n<li><strong>Period<\/strong>: The length of one complete cycle (e.g., 2\u03c0 for sine and cosine, \u03c0 for tangent).<\/li>\n\n\n\n<li><strong>Amplitude<\/strong>: Half the distance between the maximum and minimum values (e.g., 1 for basic sine and cosine).<\/li>\n\n\n\n<li><strong>Maximum\/Minimum Values<\/strong>: The peak and trough values of the functions.<\/li>\n\n\n\n<li><strong>Asymptotes<\/strong>: Vertical lines where the function is undefined (for the tangent function).<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Transformations<\/strong>: Investigating how parameters (A, B, C, D) affect the basic graphs to sketch functions of the form <math data-latex=\"y=A\\sin (B(x+C))+D  and  y=A\\cos (B(x+C))+D\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>A<\/mi><mrow><mspace width=\"0.1667em\"><\/mspace><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>B<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mi>C<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>D<\/mi><mi>a<\/mi><mi>n<\/mi><mi>d<\/mi><mi>y<\/mi><mo>=<\/mo><mi>A<\/mi><mrow><mspace width=\"0.1667em\"><\/mspace><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>B<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mi>C<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">y=A\\sin (B(x+C))+D  and  y=A\\cos (B(x+C))+D<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><strong>Applications<\/strong>: Using trigonometric functions to model and solve practical problems involving periodic behaviour.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>&gt;&gt; For General Mathematics Students&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In General Mathematics, the focus is less on sketching and transforming the graphs and more on the application of trigonometric ratios to solve practical problems:&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Reviewing sine, cosine, and tangent as ratios of side lengths in right-angled triangles (SOH-CAH-TOA).<\/li>\n\n\n\n<li>Applying the Sine Rule and Cosine Rule to solve problems involving non-right-angled triangles.<\/li>\n\n\n\n<li>Calculating the area of a triangle using the formula Area = <math data-latex=\"\\frac{1}{2}bc\\sin (A)\"><semantics><mrow><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>b<\/mi><mi>c<\/mi><mrow><mspace width=\"0.1667em\"><\/mspace><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>A<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{1}{2}bc\\sin (A)<\/annotation><\/semantics><\/math>. <\/li>\n\n\n\n<li>Solving problems involving angles of elevation and depression and bearings.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">************************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Trigonometric identities<\/strong> are<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>equalities involving trigonometric functions that are true for every value of the variable<\/strong>. In the Queensland (QLD) Year 11 Mathematical Methods syllabus, students focus on foundational identities, primarily the <strong>Pythagorean identity<\/strong> and <strong>ratio\/reciprocal identities<\/strong>, which are used to simplify expressions and solve trigonometric equations.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Key Identities Covered in QLD Year 11.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The QLD syllabus assumes familiarity with basic trigonometry and introduces the following fundamental identities, often in the context of the unit circle and general angles:&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Pythagorean Identity:<\/strong> This is a fundamental identity derived from the Pythagorean theorem applied to the unit circle:   <math data-latex=\"\\sin ^{2}\\theta +\\cos ^{2}\\theta =1\"><semantics><mrow><msup><mi>sin<\/mi><mn>2<\/mn><\/msup><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><mi>\u03b8<\/mi><mo>+<\/mo><msup><mi>cos<\/mi><mn>2<\/mn><\/msup><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><mi>\u03b8<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\sin ^{2}\\theta +\\cos ^{2}\\theta =1<\/annotation><\/semantics><\/math><\/li>\n\n\n\n<li><strong>Ratio Identity:<\/strong> This identity defines the tangent function in terms of sine and cosine:   <math data-latex=\"\\tan \\theta =\\frac{\\sin \\theta }{\\cos \\theta }\"><semantics><mrow><mrow><mi>tan<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><mo>=<\/mo><mfrac><mrow><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><\/mrow><mrow><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\tan \\theta =\\frac{\\sin \\theta }{\\cos \\theta }<\/annotation><\/semantics><\/math><\/li>\n\n\n\n<li><strong>Reciprocal Identities:<\/strong> These define the other three trigonometric functions (cosecant, secant, and cotangent) as reciprocals of the primary functions:<math data-latex=\"\\csc \\theta =\\frac{1}{\\sin \\theta }\"><semantics><mrow><mrow><mi>csc<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\csc \\theta =\\frac{1}{\\sin \\theta }<\/annotation><\/semantics><\/math> (also written as <math data-latex=\"\\mathrm{cosec}\\&gt;\\theta\"><semantics><mrow><mrow><mtext><\/mtext><mi>cosec<\/mi><\/mrow><mspace width=\"0.2222em\"><\/mspace><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathrm{cosec}\\&gt;\\theta<\/annotation><\/semantics><\/math>),   <math data-latex=\"\\sec \\theta =\\frac{1}{\\cos \\theta }\"><semantics><mrow><mrow><mi>sec<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\sec \\theta =\\frac{1}{\\cos \\theta }<\/annotation><\/semantics><\/math> ,   <math data-latex=\"\\cot \\theta =\\frac{1}{\\tan \\theta }\"><semantics><mrow><mrow><mi>cot<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mrow><mi>tan<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\cot \\theta =\\frac{1}{\\tan \\theta }<\/annotation><\/semantics><\/math><\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Application in the QLD Syllabus<\/strong>&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In Year 11, these identities are not just for memorisation; they are crucial tools for:&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Simplifying trigonometric expressions<\/strong>.<\/li>\n\n\n\n<li><strong>Proving other, more complex identities<\/strong> (which usually involves manipulating one side of an equation to match the other, simpler side).<\/li>\n\n\n\n<li><strong>Solving trigonometric equations<\/strong> within a specified domain (e.g., <math data-latex=\"0^{\\circ }\\le \\theta \\le 360^{\\circ }\"><semantics><mrow><msup><mn>0<\/mn><mo lspace=\"0em\" rspace=\"0em\">\u2218<\/mo><\/msup><mo>\u2264<\/mo><mi>\u03b8<\/mi><mo>\u2264<\/mo><msup><mn>360<\/mn><mo lspace=\"0em\" rspace=\"0em\">\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">0^{\\circ }\\le \\theta \\le 360^{\\circ }<\/annotation><\/semantics><\/math>or <math data-latex=\"0\\le \\theta \\le 2\\pi  radians\"><semantics><mrow><mn>0<\/mn><mo>\u2264<\/mo><mi>\u03b8<\/mi><mo>\u2264<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><mi>r<\/mi><mi>a<\/mi><mi>d<\/mi><mi>i<\/mi><mi>a<\/mi><mi>n<\/mi><mi>s<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">0\\le \\theta \\le 2\\pi  radians<\/annotation><\/semantics><\/math>). This often involves converting all functions in an equation to a single function using identities.<\/li>\n\n\n\n<li><strong>Understanding symmetry properties<\/strong> of trigonometric functions (e.g., <math data-latex=\"\\sin (180^{\\circ }-\\theta )=\\sin \\theta \"><semantics><mrow><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mn>180<\/mn><mo lspace=\"0em\" rspace=\"0em\">\u2218<\/mo><\/msup><mo>\u2212<\/mo><mi>\u03b8<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\sin (180^{\\circ }-\\theta )=\\sin \\theta <\/annotation><\/semantics><\/math>) which are also considered identities in the syllabus.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">More advanced identities, such as the sum and difference of angles or double-angle formulas, are typically introduced in Year 12 Mathematical Methods or Specialist Mathematics, rather than Year 11.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">****************************************************************************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">11\u5e74\u751f\u6570\u5b66\u306e\u5fdc\u7528\u4e09\u89d2\u6cd5\u3067\u306f\u3001\u57fa\u672c\u7684\u306a\u76f4\u89d2\u4e09\u89d2\u5f62\u306e\u6bd4\uff08sin cos tan\uff09\u3092\u3001\u30e9\u30b8\u30a2\u30f3\u3001\u5358\u4f4d\u5186\u3001\u4e09\u89d2\u95a2\u6570\u306e\u30b0\u30e9\u30d5\u3001\u6052\u7b49\u5f0f\u3001\u6b63\u5f26\u30fb\u4f59\u5f26\u5b9a\u7406\u3068\u3044\u3063\u305f\u3088\u308a\u8907\u96d1\u306a\u5206\u91ce\u306b\u307e\u3067\u62e1\u5f35\u3057\u307e\u3059\u3002\u3053\u308c\u306b\u3088\u308a\u3001\u76f4\u89d2\u4e09\u89d2\u5f62\u4ee5\u5916\u306e\u4e09\u89d2\u5f62\u306e\u554f\u984c\u3092\u89e3\u3044\u305f\u308a\u3001\u9762\u7a4d\u3084\u5f27\u306e\u9577\u3055\u3092\u6c42\u3081\u305f\u308a\u3001\u5468\u671f\u7684\u306a\u6319\u52d5\u3092\u30e2\u30c7\u30eb\u5316\u3057\u305f\u308a\u3067\u304d\u308b\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002\u5358\u7d14\u306a\u4e09\u89d2\u5f62\u306e\u8fba\u3084\u89d2\u5ea6\u3092\u6c42\u3081\u308b\u3060\u3051\u3067\u306a\u304f\u3001\u5b9f\u793e\u4f1a\u3078\u306e\u5fdc\u7528\u3082\u8996\u91ce\u306b\u5165\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4e3b\u306a\u6982\u5ff5\u3092\u4ee5\u4e0b\u306b\u307e\u3068\u3081\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2022 <strong>\u76f4\u89d2\u4e09\u89d2\u5f62\u306e\u67a0\u3092\u8d85\u3048\u3066<\/strong>\uff1a\u6b63\u5f26\u5b9a\u7406\uff08\u3042\u3089\u3086\u308b\u4e09\u89d2\u5f62\u306b\u9069\u7528\uff09\u3068\u4f59\u5f26\u5b9a\u7406\uff08\u3042\u3089\u3086\u308b\u4e09\u89d2\u5f62\u306b\u9069\u7528\uff09\u3092\u7528\u3044\u3066\u3001\u672a\u77e5\u306e\u8fba\u3084\u89d2\u5ea6\u3092\u6c42\u3081\u308b\u65b9\u6cd5\u3092\u5b66\u3073\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2022 <strong>\u30e9\u30b8\u30a2\u30f3<\/strong>\uff1a\u89d2\u5ea6\u3092\u5ea6\u3060\u3051\u3067\u306a\u304f\u30e9\u30b8\u30a2\u30f3\u3067\u8868\u3057\u3001\u305d\u308c\u3089\u306e\u5909\u63db\u65b9\u6cd5\u3092\u7406\u89e3\u3057\u307e\u3059\u3002\u3053\u308c\u306f\u5fae\u7a4d\u5206\u3084\u9ad8\u5ea6\u306a\u95a2\u6570\u306b\u4e0d\u53ef\u6b20\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2022 <strong>\u5358\u4f4d\u5186\u3068\u4e00\u822c\u89d2<\/strong>\uff1a\u5358\u4f4d\u5186\u3092\u7528\u3044\u3066\u3001sin\u3001cos\u3001tan\u3092\u4efb\u610f\u306e\u89d2\u5ea6\uff08\u6b63\u3001\u8ca0\u3001360\u00b0\u4ee5\u4e0a\uff09\u306b\u62e1\u5f35\u3057\u3001ASTC\uff08All Students Take Calculus\uff09\u3092\u7406\u89e3\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2022 <strong>\u4e09\u89d2\u95a2\u6570\u306e\u30b0\u30e9\u30d5<\/strong>\uff1ay = sin(x)\u3001y = cos(x)\u3001y = tan(x)\u306e\u30b0\u30e9\u30d5\u3092\u63cf\u304d\u3001\u5909\u63db\uff08\u632f\u5e45\u3001\u5468\u671f\u3001\u4f4d\u76f8\u30b7\u30d5\u30c8\uff09\u3092\u691c\u8a0e\u3057\u3066\u3001\u6ce2\u3001\u97f3\u3001\u632f\u52d5\u3092\u30e2\u30c7\u30eb\u5316\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2022 <strong>\u4e09\u89d2\u95a2\u6570\u306e\u6052\u7b49\u5f0f<\/strong>\uff1a<math data-latex=\"\\sin ^{2}\\theta +\\cos ^{2}\\theta =1\"><semantics><mrow><msup><mi>sin<\/mi><mn>2<\/mn><\/msup><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><mi>\u03b8<\/mi><mo>+<\/mo><msup><mi>cos<\/mi><mn>2<\/mn><\/msup><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><mi>\u03b8<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\sin ^{2}\\theta +\\cos ^{2}\\theta =1<\/annotation><\/semantics><\/math> \u306e\u3088\u3046\u306a\u6052\u7b49\u5f0f\u3092\u8a3c\u660e\u3057\u3001\u7528\u3044\u3066\u8907\u96d1\u306a\u5f0f\u3092\u7c21\u7565\u5316\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2022 <strong>\u5fdc\u7528<\/strong>\uff1a\u6247\u5f62\u3001\u5f27\u306e\u9577\u3055\u3001\u65b9\u4f4d\u3001\u822a\u884c\u3001\u305d\u3057\u3066\u3088\u308a\u8907\u96d1\u306a\u5e7e\u4f55\u5b66\u7684\u30b7\u30ca\u30ea\u30aa\u3092\u542b\u3080\u554f\u984c\u3092\u89e3\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u57fa\u672c\u7684\u306b\u300111\u5e74\u751f\u306e\u5fdc\u7528\u4e09\u89d2\u6cd5\u306f\u3001\u5468\u671f\u73fe\u8c61\u3092\u8a18\u8ff0\u3057\u3001\u5e7e\u4f55\u5b66\u3068\u7269\u7406\u5b66\u306b\u304a\u3051\u308b\u8907\u96d1\u306a\u6e2c\u5b9a\u554f\u984c\u3092\u89e3\u6c7a\u3059\u308b\u305f\u3081\u306e\u5f37\u529b\u306a\u30c4\u30fc\u30eb\u30ad\u30c3\u30c8\u3092\u69cb\u7bc9\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">*************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u6b63\u5f26\u5b9a\u7406<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6b63\u5f26\u5b9a\u7406\uff08\u307e\u305f\u306f\u6b63\u5f26\u5b9a\u7406\uff09\u306f\u3001\u3042\u3089\u3086\u308b\u4e09\u89d2\u5f62\u306b\u304a\u3044\u3066\u3001\u5bfe\u5fdc\u3059\u308b\u89d2\u5ea6\u304c\u3042\u308b\u5834\u5408\u306b\u672a\u77e5\u306e\u8fba\u3092\u6c42\u3081\u308b\u305f\u3081\u3001\u307e\u305f\u306f\u5bfe\u5fdc\u3059\u308b\u8fba\u3068\u5225\u306e\u89d2\u5ea6\u304c\u3042\u308b\u5834\u5408\u306b\u672a\u77e5\u306e\u89d2\u5ea6\u3092\u6c42\u3081\u308b\u305f\u3081\u306b\u4f7f\u7528\u3055\u308c\u307e\u3059\u3002\u3053\u308c\u306f\u3001\u76f4\u89d2\u3067\u306a\u3044\u4e09\u89d2\u5f62\u306e\u8fba\u3068\u305d\u308c\u3089\u306e\u5bfe\u89d2\u306e\u6b63\u5f26\u3068\u306e\u95a2\u4fc2\u3092\u78ba\u7acb\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5f0f\u306f\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u307e\u3059\u3002<math data-latex=\"\\frac{a}{\\sin A}=\\frac{b}{\\sin B}=\\frac{c}{\\sin C}\"><semantics><mrow><mfrac><mi>a<\/mi><mrow><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>A<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mi>b<\/mi><mrow><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>B<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mi>c<\/mi><mrow><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>C<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{a}{\\sin A}=\\frac{b}{\\sin B}=\\frac{c}{\\sin C}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u3053\u3067\u3001a\u3001b\u3001c \u306f\u305d\u308c\u305e\u308c\u89d2\u5ea6 A\u3001B\u3001C \u306e\u53cd\u5bfe\u5074\u306e\u8fba\u306e\u9577\u3055\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u4f59\u5f26\u5b9a\u7406<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4f59\u5f26\u5b9a\u7406\uff08\u307e\u305f\u306f\u4f59\u5f26\u5b9a\u7406\uff09\u306f\u3001\u30d4\u30bf\u30b4\u30e9\u30b9\u306e\u5b9a\u7406\u3092\u3042\u3089\u3086\u308b\u4e09\u89d2\u5f62\u306b\u4e00\u822c\u5316\u3057\u305f\u3082\u306e\u3067\u3059\u3002\u3053\u306e\u5f0f\u306f\u4e3b\u306b\u30012\u8fba\u3068\u5185\u89d2\u304c\u308f\u304b\u3063\u3066\u3044\u308b\u4e09\u89d2\u5f62\u306e3\u8fba\u76ee\u3092\u6c42\u3081\u308b\uff08SAS\uff09\u3001\u307e\u305f\u306f3\u8fba\u3059\u3079\u3066\u304c\u308f\u304b\u3063\u3066\u3044\u308b\u4e09\u89d2\u5f62\u306e\u89d2\u5ea6\u3092\u6c42\u3081\u308b\uff08SSS\uff09\u305f\u3081\u306b\u4f7f\u7528\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u306e\u5f0f\u306f\u3001\u6c42\u3081\u305f\u3044\u8fba\u306b\u5fdc\u3058\u3066\u3001\u4ee5\u4e0b\u306e3\u3064\u306e\u5f62\u5f0f\u3067\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math data-latex=\"a^{2}=b^{2}+c^{2}-2bc\\cos A\"><semantics><mrow><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>c<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>2<\/mn><mi>b<\/mi><mi>c<\/mi><mrow><mspace width=\"0.1667em\"><\/mspace><mi>cos<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a^{2}=b^{2}+c^{2}-2bc\\cos A<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math data-latex=\"b^{2}=a^{2}+c^{2}-2ac\\cos B\"><semantics><mrow><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>c<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>2<\/mn><mi>a<\/mi><mi>c<\/mi><mrow><mspace width=\"0.1667em\"><\/mspace><mi>cos<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">b^{2}=a^{2}+c^{2}-2ac\\cos B<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math data-latex=\"c^{2}=a^{2}+b^{2}-2ab\\cos C\"><semantics><mrow><msup><mi>c<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>2<\/mn><mi>a<\/mi><mi>b<\/mi><mrow><mspace width=\"0.1667em\"><\/mspace><mi>cos<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">c^{2}=a^{2}+b^{2}-2ab\\cos C<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u307e\u305f\u3001\u89d2\u5ea6\u3092\u6c42\u3081\u308b\u3088\u3046\u306b\u5909\u5f62\u3059\u308b\u3053\u3068\u3082\u3067\u304d\u307e\u3059\u3002\u4f8b\u3048\u3070\u3001\u89d2\u5ea6  <math data-latex=\"A\"><semantics><mi>A<\/mi><annotation encoding=\"application\/x-tex\">A<\/annotation><\/semantics><\/math>:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math data-latex=\"\\cos A=\\frac{b^{2}+c^{2}-a^{2}}{2bc}\"><semantics><mrow><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>A<\/mi><mo>=<\/mo><mfrac><mrow><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>c<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><msup><mi>a<\/mi><mn>2<\/mn><\/msup><\/mrow><mrow><mn>2<\/mn><mi>b<\/mi><mi>c<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\cos A=\\frac{b^{2}+c^{2}-a^{2}}{2bc}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">*****************************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u30e9\u30b8\u30a2\u30f3\uff08<\/strong><strong>\u5f27\u5ea6\u6cd5\uff09<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">QLD 11\u5e74\u751f\u306e\u6570\u5b66\uff08\u6570\u5b66\u7684\u65b9\u6cd5\uff09\u3067\u306f\u3001\u30e9\u30b8\u30a2\u30f3\u306f\u89d2\u5ea6\u3092\u6e2c\u308b\u6a19\u6e96\u7684\u306a\u65b9\u6cd5\u3067\u3042\u308a\u3001\u5f27\u306e\u9577\u3055\u304c\u534a\u5f84\u3068\u7b49\u3057\u3044\u89d2\u5ea6\u3068\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002\u3002\u5186\u306f2\u03c0\u30e9\u30b8\u30a2\u30f3\uff08360\u00b0\uff09\u306a\u306e\u3067\u3001\u03c0\u30e9\u30b8\u30a2\u30f3\u306f180\u00b0\u306b\u7b49\u3057\u304f\u306a\u308a\u307e\u3059\u3002\u30e9\u30b8\u30a2\u30f3\u306f\u5fae\u7a4d\u5206\uff08\u4e09\u89d2\u95a2\u6570\u306e\u5fae\u5206\uff09\u306b\u304a\u3044\u3066\u975e\u5e38\u306b\u91cd\u8981\u3067\u3042\u308a\u3001\u5ea6\u3067\u306f\u306a\u304f\u03c0\uff08\u7d043.14\uff09\u3092\u4f7f\u7528\u3057\u307e\u3059\u3002\u30e9\u30b8\u30a2\u30f3\u3092\u6c42\u3081\u308b\u306b\u306f\u5ea6\u306b <math data-latex=\"\\frac{\\pi }{180}\"><semantics><mfrac><mi>\u03c0<\/mi><mn>180<\/mn><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac{\\pi }{180}<\/annotation><\/semantics><\/math> \u3092\u639b\u3051\u308b\u306a\u3069\u306e\u5909\u63db\u304c\u5fc5\u8981\u3067\u3059\u3002\u307e\u305f\u3001\u5186\u95a2\u6570\u306e\u5358\u4f4d\u5186\u306b\u304a\u3044\u3066\u3082\u30e9\u30b8\u30a2\u30f3\u304c\u4f7f\u7528\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u4e3b\u8981\u6982\u5ff5<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2022 <strong>\u5b9a\u7fa9<\/strong>\uff1a1\u30e9\u30b8\u30a2\u30f3\u3068\u306f\u3001\u5186\u5f27\u306e\u9577\u3055\u304c\u534a\u5f84\u3068\u6b63\u78ba\u306b\u4e00\u81f4\u3059\u308b\u3068\u304d\u306e\u5186\u306e\u4e2d\u5fc3\u306b\u304a\u3051\u308b\u89d2\u5ea6\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2022 <strong>\u03c0 <\/strong><strong>\u3068\u306e\u95a2\u4fc2<\/strong>\uff1a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u25e6 2\u03c0 \u30e9\u30b8\u30a2\u30f3 = 360\u00b0\uff08\u5186\uff09<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u25e6 \u03c0 \u30e9\u30b8\u30a2\u30f3 = 180\u00b0\uff08\u76f4\u7dda\/\u534a\u5186\uff09<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u25e6 <math data-latex=\"\\frac{\\pi }{2}\"><semantics><mfrac><mi>\u03c0<\/mi><mn>2<\/mn><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac{\\pi }{2}<\/annotation><\/semantics><\/math> \u30e9\u30b8\u30a2\u30f3 = 90\u00b0\uff08\u76f4\u89d2\uff09<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2022 <strong>\u306a\u305c\u30e9\u30b8\u30a2\u30f3\uff1f<\/strong>\uff1a\u30e9\u30b8\u30a2\u30f3\u306f\u6570\u5f0f\u3092\u7c21\u7565\u5316\u3057\u3001\u9ad8\u7b49\u6570\u5b66\u3001\u7279\u306b\u5fae\u7a4d\u5206\u3084\u5186\u95a2\u6570\uff08\u4e09\u89d2\u6cd5\uff09\u3092\u6271\u3046\u969b\u306b\u4e0d\u53ef\u6b20\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2022 <strong>\u5358\u4f4d<\/strong>\uff1a\u30e9\u30b8\u30a2\u30f3\u306b\u306f\u901a\u5e38\u8a18\u53f7\u306f\u3042\u308a\u307e\u305b\u3093\u304c\u3001\u5c0f\u6587\u5b57\u306e\u300cc\u300d\uff08\u5186\uff09\u307e\u305f\u306f\u300crad\u300d\u304c\u4f7f\u308f\u308c\u308b\u3053\u3068\u3082\u3042\u308a\u307e\u3059\u3002\u305f\u3060\u3057\u3001\u6570\u5b66\u7684\u306a\u6587\u8108\u3067\u306f\u3001\u8a18\u53f7\u304c\u306a\u304f\u3066\u3082\u30e9\u30b8\u30a2\u30f3\u304c\u610f\u5473\u3092\u6301\u3064\u3053\u3068\u306f\u3088\u304f\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5909\u63db\u5f0f<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2022 <strong>\u5ea6\u304b\u3089\u30e9\u30b8\u30a2\u30f3\u3078<\/strong>\uff1a\u89d2\u5ea6\uff08\u5ea6\uff09\u306b  <math data-latex=\"\\frac{\\pi }{180} \"><semantics><mfrac><mi>\u03c0<\/mi><mn>180<\/mn><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac{\\pi }{180} <\/annotation><\/semantics><\/math>  \u3092\u639b\u3051\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2022 <strong>\u30e9\u30b8\u30a2\u30f3\u304b\u3089\u5ea6\u3078<\/strong>\uff1a\u89d2\u5ea6\uff08\u30e9\u30b8\u30a2\u30f3\uff09\u306b <math data-latex=\"\\frac {180}{\\pi }\"><semantics><mfrac><mn>180<\/mn><mi>\u03c0<\/mi><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac {180}{\\pi }<\/annotation><\/semantics><\/math>  \u3092\u639b\u3051\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u5b9f\u8df5\u7de8<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2022 <strong>\u8a08\u7b97\u6a5f\u30e2\u30fc\u30c9<\/strong>\uff1a\u30e9\u30b8\u30a2\u30f3\u3092\u6271\u3046\u969b\u306f\u3001\u8a08\u7b97\u6a5f\u304c\u300cRAD\u300d\u30e2\u30fc\u30c9\u306b\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3057\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2022 <strong>\u5358\u4f4d\u5186<\/strong>: \u5358\u4f4d\u5186\uff1a\u534a\u5f84\u304c\uff11\u306e\u5186 ( 1 \u30e9\u30b8\u30a2\u30b9) \u4e0a\u306e\u89d2\u5ea6\u306f\u30e9\u30b8\u30a2\u30f3\u3067\u7c21\u5358\u306b\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059 (x \u8ef8\u306e\u5024\u306f \u03c0\u30012\u03c0 \u306a\u3069\u306b\u306a\u308a\u307e\u3059)\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u534a\u5f84\u306e\u9577\u3055\u304c1\u306e\u5186\u3092<strong>\u5358\u4f4d\u5186<\/strong>\u3068\u3044\u3046\uff0e<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u901a\u5e38\u00a0\u5ea7\u6a19\u5e73\u9762\u4e0a\u306e\u539f\u70b9\u306b\u5186\u306e\u4e2d\u5fc3\u3092\u3068\u308a<strong>\u5358\u4f4d\u5186<\/strong>\u3092\u63cf\u304f(\u53f3\u56f3\u53c2\u7167)\uff0e<strong>\u5358\u4f4d\u5186<\/strong>\u4e0a\u306e\u70b9\u306e\u5ea7\u6a19\u00a0\u00a0\u306f\uff0c\u8ef8\u3068\u7dda\u5206\u306e<a href=\"https:\/\/w3e.kanazawa-it.ac.jp\/math\/category\/sankakukansuu\/sankakuhi\/henkan-tex.cgi?target=\/math\/category\/sankakukansuu\/sankakuhi\/kakudo-no-teigi.html\">\u306a\u3059\u89d2<\/a>\u3092\u56f3\u306e\u3088\u3046\u306b\u00a0\u3068\u3059\u308b\u3068\uff0c\u4e09\u89d2\u95a2\u6570\u306e\u5b9a\u7fa9\u3088\u308a<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"278\" height=\"279\" src=\"https:\/\/archive4ones.com\/2ndstudy\/wp-content\/uploads\/sites\/8\/2026\/01\/Unit-Circle.png\" alt=\"\" class=\"wp-image-1071\" srcset=\"https:\/\/archive4ones.com\/2ndstudy\/wp-content\/uploads\/sites\/8\/2026\/01\/Unit-Circle.png 278w, https:\/\/archive4ones.com\/2ndstudy\/wp-content\/uploads\/sites\/8\/2026\/01\/Unit-Circle-150x150.png 150w\" sizes=\"auto, (max-width: 278px) 100vw, 278px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u3068\u306a\u308b\uff0e<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">*************************************************************************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u4e09\u89d2\u30b0\u30e9\u30d5<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30af\u30a4\u30fc\u30f3\u30ba\u30e9\u30f3\u30c9\u5dde\u306e11\u5e74\u751f\u6570\u5b66\u3067\u306f\u3001\u4e09\u89d2\u30b0\u30e9\u30d5\u306f\u95a2\u6570\u3068\u95a2\u4fc2\u306b\u9069\u7528\u3055\u308c\u308b\u6570\u5b66\u7684\u624b\u6cd5\u3067\u3059\u3002\u6d77\u306e\u6f6e\u6c50\u3084\u632f\u308a\u5b50\u306e\u63fa\u308c\u3068\u3044\u3063\u305f\u3001\u5468\u671f\u7684\u307e\u305f\u306f\u5468\u671f\u7684\u306a\u73fe\u5b9f\u4e16\u754c\u306e\u73fe\u8c61\u3092\u30e2\u30c7\u30eb\u5316\u3059\u308b\u305f\u3081\u306b\u7528\u3044\u3089\u308c\u307e\u3059\u3002\u4e00\u822c\u6570\u5b66\u3067\u306f\u3001\u30b0\u30e9\u30d5\u81ea\u4f53\u306e\u63cf\u753b\u3084\u5909\u5f62\u3088\u308a\u3082\u3001\u76f4\u89d2\u4e09\u89d2\u5f62\u3068\u975e\u76f4\u89d2\u4e09\u89d2\u5f62\u306b\u304a\u3051\u308b\u4e09\u89d2\u6cd5\u306e\u5fdc\u7528\u306b\u91cd\u70b9\u304c\u7f6e\u304b\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&gt;&gt; <strong>\u6570\u5b66\u7684\u624b\u6cd5\u3092\u5b66\u3076\u751f\u5f92\u306e\u65b9\u3078<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">11\u5e74\u751f\u6570\u5b66\u7684\u624b\u6cd5\u306e\u30b7\u30e9\u30d0\u30b9\u3067\u306f\u3001\u4e09\u89d2\u95a2\u6570\u3068\u305d\u308c\u306b\u5bfe\u5fdc\u3059\u308b\u30b0\u30e9\u30d5\u306b\u3064\u3044\u3066\u3001\u4ee5\u4e0b\u306e\u5185\u5bb9\u3092\u542b\u3081\u3066\u8a73\u7d30\u306b\u5b66\u7fd2\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2022 <strong>\u5358\u4f4d\u5186\u306e\u7406\u89e3<\/strong>\uff1a\u5358\u4f4d\u5186\u3092\u7528\u3044\u3066\u6b63\u5f26\uff08sin <math data-latex=\"\\theta\"><semantics><mi>\u03b8<\/mi><annotation encoding=\"application\/x-tex\">\\theta<\/annotation><\/semantics><\/math>\uff09\u3001\u4f59\u5f26\uff08cos <math data-latex=\"\\theta\"><semantics><mi>\u03b8<\/mi><annotation encoding=\"application\/x-tex\">\\theta<\/annotation><\/semantics><\/math>\uff09\u3001\u6b63\u63a5\uff08tan <math data-latex=\"\\theta\"><semantics><mi>\u03b8<\/mi><annotation encoding=\"application\/x-tex\">\\theta<\/annotation><\/semantics><\/math>\uff09\u95a2\u6570\u3092\u5b9a\u7fa9\u3057\u3001\u305d\u308c\u3089\u306e\u5468\u671f\u6027\u3092\u7406\u89e3\u3059\u308b\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2022 <strong>\u89d2\u5ea6\u306e\u6e2c\u5b9a<\/strong>\uff1a\u5ea6\u3068\u30e9\u30b8\u30a2\u30f3\u306e\u4e21\u65b9\u3092\u4f7f\u7528\u3057\u3001\u305d\u308c\u3089\u306e\u5909\u63db\u3092\u884c\u3044\u307e\u3059\uff08\u4f8b\uff1a360\u00b0 = 2\u03c0 \u30e9\u30b8\u30a2\u30f3\uff09\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2022 <strong>\u57fa\u672c\u30b0\u30e9\u30d5\u306e\u63cf\u753b<\/strong>\uff1a\u62e1\u5f35\u3055\u308c\u305f\u5b9a\u7fa9\u57df\u306b\u304a\u3051\u308b <math data-latex=\"y = sin(x)\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>s<\/mi><mi>i<\/mi><mi>n<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">y = sin(x)<\/annotation><\/semantics><\/math> \u3001<math data-latex=\"y = cos(x)\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>c<\/mi><mi>o<\/mi><mi>s<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">y = cos(x)<\/annotation><\/semantics><\/math> and <math data-latex=\"y = tan(x) \"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>t<\/mi><mi>a<\/mi><mi>n<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">y = tan(x) <\/annotation><\/semantics><\/math>\u306e\u30b0\u30e9\u30d5\u3092\u8a8d\u8b58\u3057\u3001\u63cf\u753b\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2022 <strong>\u4e3b\u306a\u7279\u5fb4<\/strong>\uff1a\u3053\u308c\u3089\u306e\u30b0\u30e9\u30d5\u306e\u7279\u6027\u3092\u7279\u5b9a\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u25e6 \u5468\u671f\uff1a1\u5468\u671f\u306e\u9577\u3055\uff08\u4f8b\uff1a\u6b63\u5f26\u95a2\u6570\u3068\u4f59\u5f26\u95a2\u6570\u306e\u5834\u5408\u306f2\u03c0\u3001\u6b63\u63a5\u95a2\u6570\u306e\u5834\u5408\u306f\u03c0\uff09\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u25e6 \u632f\u5e45\uff1a\u6700\u5927\u5024\u3068\u6700\u5c0f\u5024\u306e\u9593\u306e\u8ddd\u96e2\u306e\u534a\u5206\uff08\u4f8b\uff1a\u57fa\u672c\u7684\u306a\u6b63\u5f26\u95a2\u6570\u3068\u4f59\u5f26\u95a2\u6570\u306e\u5834\u5408\u306f1\uff09\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u25e6 <strong>\u6700\u5927\u5024<\/strong><strong>\/<\/strong><strong>\u6700\u5c0f\u5024<\/strong>\uff1a\u95a2\u6570\u306e\u30d4\u30fc\u30af\u5024\u3068\u8c37\u5024\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u25e6 <strong>\u6f38\u8fd1\u7dda<\/strong>\uff1a\u95a2\u6570\u304c\u5b9a\u7fa9\u3055\u308c\u3066\u3044\u306a\u3044\u5782\u76f4\u7dda\uff08\u6b63\u63a5\u95a2\u6570\u306e\u5834\u5408\uff09\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2022 <strong>\u5909\u63db<\/strong>\uff1a\u30d1\u30e9\u30e1\u30fc\u30bf\uff08A\u3001B\u3001C\u3001D\uff09\u304c\u57fa\u672c\u30b0\u30e9\u30d5\u306b\u3069\u306e\u3088\u3046\u306b\u5f71\u97ff\u3059\u308b\u304b\u3092\u8abf\u3079\u3001<math data-latex=\"y=A\\sin (B(x+C))+D and y=A\\cos (B(x+C))+D\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>A<\/mi><mrow><mspace width=\"0.1667em\"><\/mspace><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>B<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mi>C<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>D<\/mi><mi>a<\/mi><mi>n<\/mi><mi>d<\/mi><mi>y<\/mi><mo>=<\/mo><mi>A<\/mi><mrow><mspace width=\"0.1667em\"><\/mspace><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>B<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mi>C<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">y=A\\sin (B(x+C))+D and y=A\\cos (B(x+C))+D<\/annotation><\/semantics><\/math> \u306e\u5f62\u306e\u95a2\u6570\u3092\u63cf\u753b\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2022 <strong>\u5fdc\u7528<\/strong>\uff1a\u4e09\u89d2\u95a2\u6570\u3092\u7528\u3044\u3066\u3001\u5468\u671f\u7684\u306a\u6319\u52d5\u3092\u4f34\u3046\u5b9f\u7528\u7684\u306a\u554f\u984c\u3092\u30e2\u30c7\u30eb\u5316\u3057\u3001\u89e3\u304f\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>&gt;&gt; <\/strong><strong>\u4e00\u822c\u6570\u5b66\u3092\u5b66\u3076\u5b66\u751f\u5411\u3051<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4e00\u822c\u6570\u5b66\u3067\u306f\u3001\u30b0\u30e9\u30d5\u306e\u63cf\u753b\u3084\u5909\u63db\u3088\u308a\u3082\u3001\u4e09\u89d2\u6bd4\u3092\u5fdc\u7528\u3057\u3066\u5b9f\u7528\u7684\u306a\u554f\u984c\u3092\u89e3\u304f\u3053\u3068\u306b\u91cd\u70b9\u304c\u7f6e\u304b\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2022 \u76f4\u89d2\u4e09\u89d2\u5f62\u306e\u8fba\u306e\u9577\u3055\u306e\u6bd4\u3068\u3057\u3066\u306e\u6b63\u5f26\u3001\u4f59\u5f26\u3001\u6b63\u63a5\uff08SOH-CAH-TOA\uff09\u3092\u5fa9\u7fd2\u3059\u308b\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2022 \u6b63\u5f26\u5b9a\u7406\u3068\u4f59\u5f26\u5b9a\u7406\u3092\u7528\u3044\u3066\u3001\u76f4\u89d2\u3067\u306a\u3044\u4e09\u89d2\u5f62\u306e\u554f\u984c\u3092\u89e3\u304f\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2022 \u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u3092\u3001Area = <math data-latex=\"\\frac{1}{2}bc\\sin (A)\"><semantics><mrow><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>b<\/mi><mi>c<\/mi><mrow><mspace width=\"0.1667em\"><\/mspace><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>A<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{1}{2}bc\\sin (A)<\/annotation><\/semantics><\/math>  \u3068\u3044\u3046\u516c\u5f0f\u3092\u7528\u3044\u3066\u8a08\u7b97\u3059\u308b\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2022 \u4ef0\u89d2\u3001\u4fef\u89d2\u3001\u65b9\u4f4d\u89d2\u3092\u542b\u3080\u554f\u984c\u3092\u89e3\u304f\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">**********************************************************************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u4e09\u89d2\u95a2\u6570\u306e\u6052\u7b49\u5f0f<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4e09\u89d2\u95a2\u6570\u306e\u6052\u7b49\u5f0f\u306f\u3001\u5909\u6570\u306e\u3042\u3089\u3086\u308b\u5024\u306b\u5bfe\u3057\u3066\u6210\u308a\u7acb\u3064\u4e09\u89d2\u95a2\u6570\u306e\u7b49\u5f0f\u3067\u3059\u3002\u30af\u30a4\u30fc\u30f3\u30ba\u30e9\u30f3\u30c9\u5dde\uff08QLD\uff09\u306e11\u5e74\u751f\u6570\u5b66\u6307\u5c0e\u6cd5\u306e\u30b7\u30e9\u30d0\u30b9\u3067\u306f\u3001\u751f\u5f92\u306f\u57fa\u790e\u7684\u306a\u6052\u7b49\u5f0f\u3001\u7279\u306b\u30d4\u30bf\u30b4\u30e9\u30b9\u306e\u6052\u7b49\u5f0f\u3068\u6bd4\uff0f\u9006\u6570\u306e\u6052\u7b49\u5f0f\u306b\u91cd\u70b9\u7684\u306b\u53d6\u308a\u7d44\u307f\u307e\u3059\u3002\u3053\u308c\u3089\u306e\u6052\u7b49\u5f0f\u306f\u3001\u5f0f\u306e\u7c21\u7565\u5316\u3084\u4e09\u89d2\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5\u306b\u7528\u3044\u3089\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">QLD 11\u5e74\u751f\u3067\u6271\u3046\u4e3b\u8981\u306a\u6052\u7b49\u5f0f<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">QLD\u306e\u30b7\u30e9\u30d0\u30b9\u3067\u306f\u3001\u57fa\u672c\u7684\u306a\u4e09\u89d2\u6cd5\u306e\u77e5\u8b58\u3092\u524d\u63d0\u3068\u3057\u3001\u5358\u4f4d\u5186\u3084\u4e00\u822c\u7684\u306a\u89d2\u5ea6\u306e\u6587\u8108\u3067\u3001\u4ee5\u4e0b\u306e\u57fa\u672c\u7684\u306a\u6052\u7b49\u5f0f\u3092\u5c0e\u5165\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30d4\u30bf\u30b4\u30e9\u30b9\u306e\u5b9a\u7406\uff1a\u3053\u308c\u306f\u3001\u30d4\u30bf\u30b4\u30e9\u30b9\u306e\u5b9a\u7406\u3092\u5358\u4f4d\u5186\u306b\u9069\u7528\u3059\u308b\u3053\u3068\u3067\u5f97\u3089\u308c\u308b\u57fa\u672c\u7684\u306a\u6052\u7b49\u5f0f\u3067\u3059\u3002<math data-latex=\"\\sin ^{2}\\theta + \\cos ^{2}\\theta =1\"><semantics><mrow><msup><mi>sin<\/mi><mn>2<\/mn><\/msup><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><mi>\u03b8<\/mi><mo>+<\/mo><msup><mi>cos<\/mi><mn>2<\/mn><\/msup><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><mi>\u03b8<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\sin ^{2}\\theta + \\cos ^{2}\\theta =1<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6bd4\u306e\u6052\u7b49\u5f0f\uff1a\u3053\u306e\u6052\u7b49\u5f0f\u306f\u3001\u6b63\u5f26\u95a2\u6570\u3068\u4f59\u5f26\u95a2\u6570\u3092\u7528\u3044\u3066\u6b63\u63a5\u95a2\u6570\u3092\u5b9a\u7fa9\u3057\u307e\u3059\u3002<math data-latex=\"\\tan \\theta =\\frac{\\sin \\theta }{\\cos \\theta }\"><semantics><mrow><mrow><mi>tan<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><mo>=<\/mo><mfrac><mrow><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><\/mrow><mrow><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\tan \\theta =\\frac{\\sin \\theta }{\\cos \\theta }<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u9006\u6570\u306e\u6052\u7b49\u5f0f\uff1a\u3053\u308c\u3089\u306f\u3001\u4ed6\u306e3\u3064\u306e\u4e09\u89d2\u95a2\u6570\uff08\u30b3\u30bb\u30ab\u30f3\u30c8\u3001\u30bb\u30ab\u30f3\u30c8\u3001\u30b3\u30bf\u30f3\u30b8\u30a7\u30f3\u30c8\uff09\u3092\u3001\u57fa\u672c\u95a2\u6570\u306e\u9006\u6570\u3068\u3057\u3066\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math data-latex=\"\\csc \\theta =\\frac{1}{\\sin \\theta }\"><semantics><mrow><mrow><mi>csc<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\csc \\theta =\\frac{1}{\\sin \\theta }<\/annotation><\/semantics><\/math>\uff08 <math data-latex=\"\\mathrm{cosec}\"><semantics><mrow><mtext><\/mtext><mi>cosec<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathrm{cosec}<\/annotation><\/semantics><\/math>  \u3068\u3082\u8868\u8a18\u3055\u308c\u307e\u3059\uff09<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math data-latex=\"\\sec \\theta =\\frac{1}{\\cos \\theta }\"><semantics><mrow><mrow><mi>sec<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\sec \\theta =\\frac{1}{\\cos \\theta }<\/annotation><\/semantics><\/math>    <\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math data-latex=\"\\cot \\theta =\\frac{1}{\\tan \\theta }\"><semantics><mrow><mrow><mi>cot<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mrow><mi>tan<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\cot \\theta =\\frac{1}{\\tan \\theta }<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">QLD\u30b7\u30e9\u30d0\u30b9\u306b\u304a\u3051\u308b\u5fdc\u7528<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">11\u5e74\u751f\u3067\u306f\u3001\u3053\u308c\u3089\u306e\u6052\u7b49\u5f0f\u306f\u5358\u306a\u308b\u6697\u8a18\u3067\u306f\u306a\u304f\u3001\u4ee5\u4e0b\u306e\u305f\u3081\u306e\u91cd\u8981\u306a\u30c4\u30fc\u30eb\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4e09\u89d2\u95a2\u6570\u306e\u5f0f\u3092\u7c21\u7565\u5316\u3059\u308b\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3088\u308a\u8907\u96d1\u306a\u6052\u7b49\u5f0f\u306e\u8a3c\u660e\uff08\u901a\u5e38\u306f\u3001\u65b9\u7a0b\u5f0f\u306e\u7247\u8fba\u3092\u3082\u3046\u4e00\u65b9\u306e\u3088\u308a\u5358\u7d14\u306a\u8fba\u3068\u4e00\u81f4\u3059\u308b\u3088\u3046\u306b\u64cd\u4f5c\u3059\u308b\uff09\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6307\u5b9a\u3055\u308c\u305f\u9818\u57df\uff08\u4f8b\uff1a<math data-latex=\"0^{\\circ }\\le \\theta \\le 360\u200b\u200b^{\\circ }\"><semantics><mrow><msup><mn>0<\/mn><mo lspace=\"0em\" rspace=\"0em\">\u2218<\/mo><\/msup><mo>\u2264<\/mo><mi>\u03b8<\/mi><mo>\u2264<\/mo><mn>360<\/mn><mtext>\u200b<\/mtext><msup><mtext>\u200b<\/mtext><mo lspace=\"0em\" rspace=\"0em\">\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">0^{\\circ }\\le \\theta \\le 360\u200b\u200b^{\\circ }<\/annotation><\/semantics><\/math> \u307e\u305f\u306f <math data-latex=\"0\\le \\theta \\le 2\\pi \"><semantics><mrow><mn>0<\/mn><mo>\u2264<\/mo><mi>\u03b8<\/mi><mo>\u2264<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">0\\le \\theta \\le 2\\pi <\/annotation><\/semantics><\/math>  \u30e9\u30b8\u30a2\u30f3\uff09\u5185\u3067\u4e09\u89d2\u95a2\u6570\u306e\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3002\u3053\u308c\u306b\u306f\u3001\u591a\u304f\u306e\u5834\u5408\u3001\u65b9\u7a0b\u5f0f\u5185\u306e\u3059\u3079\u3066\u306e\u95a2\u6570\u3092\u6052\u7b49\u5f0f\u3092\u7528\u3044\u3066\u5358\u4e00\u306e\u95a2\u6570\u306b\u5909\u63db\u3059\u308b\u3053\u3068\u304c\u542b\u307e\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30b7\u30e9\u30d0\u30b9\u3067\u306f\u6052\u7b49\u5f0f\u3068\u307f\u306a\u3055\u308c\u308b\u4e09\u89d2\u95a2\u6570\uff08\u4f8b\uff1a<math data-latex=\"\\sin (180^{\\circ }-\\theta )=\\sin \\theta\"><semantics><mrow><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mn>180<\/mn><mo lspace=\"0em\" rspace=\"0em\">\u2218<\/mo><\/msup><mo>\u2212<\/mo><mi>\u03b8<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\sin (180^{\\circ }-\\theta )=\\sin \\theta<\/annotation><\/semantics><\/math>\uff09\u306e\u5bfe\u79f0\u6027\u3092\u7406\u89e3\u3059\u308b\u3053\u3068\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u89d2\u5ea6\u306e\u548c\u3068\u5dee\u3001\u500d\u89d2\u306e\u516c\u5f0f\u3068\u3044\u3063\u305f\u3088\u308a\u9ad8\u5ea6\u306a\u6052\u7b49\u5f0f\u306f\u3001\u901a\u5e38\u300111\u5e74\u751f\u3067\u306f\u306a\u304f\u300112\u5e74\u751f\u306e\u6570\u5b66\u30e1\u30bd\u30c3\u30c9\u307e\u305f\u306f\u5c02\u9580\u6570\u5b66\u3067\u5c0e\u5165\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Applied Trigonometry in Year 11 Maths extends basic right-angled triangle ratios (sin cos tan) to more complex areas like radians, unit circle, trigonometric graphs, identities, and the Sine\/Cosine Rules, allowing you to solve problems in non-right-angled triangles, find areas, arc lengths, and model periodic behaviour, moving beyond just finding sides and angles in simple triangles [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[],"class_list":["post-1069","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1069","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1069"}],"version-history":[{"count":3,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1069\/revisions"}],"predecessor-version":[{"id":1073,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1069\/revisions\/1073"}],"wp:attachment":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1069"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1069"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1069"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}