{"id":1206,"date":"2026-01-21T20:21:23","date_gmt":"2026-01-21T10:21:23","guid":{"rendered":"https:\/\/archive4ones.com\/2ndstudy\/?p=1206"},"modified":"2026-01-21T20:21:23","modified_gmt":"2026-01-21T10:21:23","slug":"year11-math-4-1-4-trigonometry-and-functions","status":"publish","type":"post","link":"https:\/\/archive4ones.com\/2ndstudy\/?p=1206","title":{"rendered":"Year11-MATH-4-1-4 Trigonometry and Functions"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">In Queensland Grade 11 Mathematics, &#8220;Trigonometry and Functions&#8221; in Unit 2 (likely for Mathematical Methods or Specialist Mathematics) involves<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><mark>extending beyond right-angled triangles to understand trigonometric functions, identities, graphs (like sine, cosine, tangent), their applications, and advanced function concepts, including potentially complex numbers and vectors, building on earlier algebra, graphing, and geometric foundations<\/mark>.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Here&#8217;s a breakdown of what it covers:&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Functions:<\/strong> Deep dives into different types of functions (linear, quadratic, exponential, etc.), functional notation, graphing, and transformations, as functions are core to modeling real-world scenarios.<\/li>\n\n\n\n<li><strong>Trigonometry:<\/strong>\n<ul class=\"wp-block-list\">\n<li><strong>Ratios &amp; Identities:<\/strong> Expanding beyond basic SOH CAH TOA to use trigonometric identities (like <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>sin<\/mi><mn>2<\/mn><\/msup><mi>\u03b8<\/mi><mo>+<\/mo><msup><mi>cos<\/mi><mn>2<\/mn><\/msup><mi>\u03b8<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"text\/plain\">sine squared theta plus cosine squared theta equals 1<\/annotation><\/semantics><\/math>) to simplify expressions and solve equations.<\/li>\n\n\n\n<li><strong>Graphs:<\/strong> Analyzing the periodic nature and graphs of sine, cosine, and tangent functions.<\/li>\n\n\n\n<li><strong>Applications:<\/strong> Using trigonometry to solve problems in geometry, physics, and other fields.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Specialist Mathematics Unit 2 Specifics (if applicable):<\/strong> If it&#8217;s Specialist Mathematics, Unit 2 might also introduce complex numbers (imaginary numbers, Argand diagrams, De Moivre&#8217;s Theorem) and vectors, as these build on function and trigonometry concepts.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">In essence, you learn how to describe and model relationships using algebraic functions and trigonometric principles, preparing you for calculus and advanced mathematics.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">***************************************************************************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Queensland Year 11 Specialist Mathematics, particularly in Unit 2, focuses on advanced trigonometry and function behavior, including <mark>sketching complex graphs, solving non-linear trigonometric equations, and applying identities<\/mark>. Key topics include manipulating (<math data-latex=\"\\sec \"><semantics><mrow><mi>sec<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><annotation encoding=\"application\/x-tex\">\\sec <\/annotation><\/semantics><\/math>), (<math data-latex=\"\\csc \"><semantics><mrow><mi>csc<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><annotation encoding=\"application\/x-tex\">\\csc <\/annotation><\/semantics><\/math>), (<math data-latex=\"\\cot \"><semantics><mrow><mi>cot<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><annotation encoding=\"application\/x-tex\">\\cot <\/annotation><\/semantics><\/math>) functions, proving identities, and analyzing graphs with phase shifts and transformations.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Sample Problems and Solutions<\/strong>&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Problem 1: Sketching Trigonometric Graphs<\/strong>\n<ul class=\"wp-block-list\">\n<li><strong>Question:<\/strong> Sketch the graph of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mn>3<\/mn><mi>sin<\/mi><mo>(<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mfrac><mi>\u03c0<\/mi><mn>2<\/mn><\/mfrac><mo>)<\/mo><mo>+<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"text\/plain\">y equals 3 sine open paren 2 x minus the fraction with numerator pi and denominator 2 end-fraction close paren plus 1<\/annotation><\/semantics><\/math> for <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>0<\/mn><mo>\u2264<\/mo><mi>x<\/mi><mo>\u2264<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><annotation encoding=\"text\/plain\">0 is less than or equal to x is less than or equal to 2 pi<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><strong>Solution:<\/strong>\n<ol class=\"wp-block-list\">\n<li><strong>Amplitude:<\/strong> 3.<\/li>\n\n\n\n<li><strong>Period:<\/strong> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mrow><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><mi>n<\/mi><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><mn>2<\/mn><\/mfrac><mo>=<\/mo><mi>\u03c0<\/mi><\/mrow><annotation encoding=\"text\/plain\">the fraction with numerator 2 pi and denominator n end-fraction equals the fraction with numerator 2 pi and denominator 2 end-fraction equals pi<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><strong>Phase Shift:<\/strong> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mo>(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mfrac><mi>\u03c0<\/mi><mn>4<\/mn><\/mfrac><mo>)<\/mo><mo>=<\/mo><mn>0<\/mn><mo>\u27f9<\/mo><mi>x<\/mi><mo>=<\/mo><mfrac><mi>\u03c0<\/mi><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"text\/plain\">2 open paren x minus the fraction with numerator pi and denominator 4 end-fraction close paren equals 0 \u27f9 x equals the fraction with numerator pi and denominator 4 end-fraction<\/annotation><\/semantics><\/math> (shifted right by <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mfrac><mi>\u03c0<\/mi><mn>4<\/mn><\/mfrac><annotation encoding=\"text\/plain\">the fraction with numerator pi and denominator 4 end-fraction<\/annotation><\/semantics><\/math>).<\/li>\n\n\n\n<li><strong>Vertical Shift:<\/strong> Up 1 unit (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mn>+1<\/mn><annotation encoding=\"text\/plain\">positive 1<\/annotation><\/semantics><\/math>).<\/li>\n\n\n\n<li><strong>Sketch:<\/strong> The graph starts 1 unit up, has a maximum of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>1<\/mn><mo>+<\/mo><mn>3<\/mn><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"text\/plain\">1 plus 3 equals 4<\/annotation><\/semantics><\/math>, and a minimum of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>1<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mo>=<\/mo><mn>-2<\/mn><\/mrow><annotation encoding=\"text\/plain\">1 minus 3 equals negative 2<\/annotation><\/semantics><\/math>. Sketch a sine curve starting at <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mi>\u03c0<\/mi><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"text\/plain\">x equals the fraction with numerator pi and denominator 4 end-fraction<\/annotation><\/semantics><\/math>, ending at <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mrow><mn>5<\/mn><mi>\u03c0<\/mi><\/mrow><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"text\/plain\">x equals the fraction with numerator 5 pi and denominator 4 end-fraction<\/annotation><\/semantics><\/math> (one period), repeated over the range.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Problem 2: Proving Trigonometric Identities<\/strong>\n<ul class=\"wp-block-list\">\n<li><strong>Question:<\/strong> Prove the identity <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mrow><mi>cos<\/mi><mi>\u03b8<\/mi><\/mrow><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>sin<\/mi><mi>\u03b8<\/mi><\/mrow><\/mfrac><mo>\u2212<\/mo><mi>tan<\/mi><mi>\u03b8<\/mi><mo>=<\/mo><mi>sec<\/mi><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"text\/plain\">the fraction with numerator cosine theta and denominator 1 minus sine theta end-fraction minus tangent theta equals secant theta<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><strong>Solution:<\/strong>\n<ol class=\"wp-block-list\">\n<li>Start with LHS: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mrow><mi>cos<\/mi><mi>\u03b8<\/mi><\/mrow><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>sin<\/mi><mi>\u03b8<\/mi><\/mrow><\/mfrac><mo>\u2212<\/mo><mfrac><mrow><mi>sin<\/mi><mi>\u03b8<\/mi><\/mrow><mrow><mi>cos<\/mi><mi>\u03b8<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"text\/plain\">the fraction with numerator cosine theta and denominator 1 minus sine theta end-fraction minus the fraction with numerator sine theta and denominator cosine theta end-fraction<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li>Common denominator: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mfrac><mrow><msup><mi>cos<\/mi><mn>2<\/mn><\/msup><mi>\u03b8<\/mi><mo>\u2212<\/mo><mi>sin<\/mi><mi>\u03b8<\/mi><mo>(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>sin<\/mi><mi>\u03b8<\/mi><mo>)<\/mo><\/mrow><mrow><mi>cos<\/mi><mi>\u03b8<\/mi><mo>(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>sin<\/mi><mi>\u03b8<\/mi><mo>)<\/mo><\/mrow><\/mfrac><annotation encoding=\"text\/plain\">the fraction with numerator cosine squared theta minus sine theta open paren 1 minus sine theta close paren and denominator cosine theta open paren 1 minus sine theta close paren end-fraction<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li>Expand: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mfrac><mrow><msup><mi>cos<\/mi><mn>2<\/mn><\/msup><mi>\u03b8<\/mi><mo>\u2212<\/mo><mi>sin<\/mi><mi>\u03b8<\/mi><mo>+<\/mo><msup><mi>sin<\/mi><mn>2<\/mn><\/msup><mi>\u03b8<\/mi><\/mrow><mrow><mi>cos<\/mi><mi>\u03b8<\/mi><mo>(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>sin<\/mi><mi>\u03b8<\/mi><mo>)<\/mo><\/mrow><\/mfrac><annotation encoding=\"text\/plain\">the fraction with numerator cosine squared theta minus sine theta plus sine squared theta and denominator cosine theta open paren 1 minus sine theta close paren end-fraction<\/annotation><\/semantics><\/math>).<\/li>\n\n\n\n<li>Use <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>cos<\/mi><mn>2<\/mn><\/msup><mi>\u03b8<\/mi><mo>+<\/mo><msup><mi>sin<\/mi><mn>2<\/mn><\/msup><mi>\u03b8<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"text\/plain\">cosine squared theta plus sine squared theta equals 1<\/annotation><\/semantics><\/math>: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mfrac><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>sin<\/mi><mi>\u03b8<\/mi><\/mrow><mrow><mi>cos<\/mi><mi>\u03b8<\/mi><mo>(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>sin<\/mi><mi>\u03b8<\/mi><mo>)<\/mo><\/mrow><\/mfrac><annotation encoding=\"text\/plain\">the fraction with numerator 1 minus sine theta and denominator cosine theta open paren 1 minus sine theta close paren end-fraction<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li>Cancel <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>sin<\/mi><mi>\u03b8<\/mi><mo>)<\/mo><\/mrow><annotation encoding=\"text\/plain\">open paren 1 minus sine theta close paren<\/annotation><\/semantics><\/math>: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mn>1<\/mn><mrow><mi>cos<\/mi><mi>\u03b8<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mi>sec<\/mi><mi>\u03b8<\/mi><mo>=<\/mo><mtext>RHS<\/mtext><\/mrow><annotation encoding=\"text\/plain\">the fraction with numerator 1 and denominator cosine theta end-fraction equals secant theta equals RHS<\/annotation><\/semantics><\/math>.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Problem 3: Solving Trigonometric Equations<\/strong>\n<ul class=\"wp-block-list\">\n<li><strong>Question:<\/strong> Solve <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msqrt><mn>3<\/mn><\/msqrt><mi>cot<\/mi><mi>\u03b8<\/mi><mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"text\/plain\">the square root of 3 end-root cotangent theta plus 1 equals 0<\/annotation><\/semantics><\/math> for <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>0<\/mn><mo>\u2264<\/mo><mi>\u03b8<\/mi><mo>\u2264<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><annotation encoding=\"text\/plain\">0 is less than or equal to theta is less than or equal to 2 pi<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><strong>Solution:<\/strong>\n<ol class=\"wp-block-list\">\n<li>Rearrange: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msqrt><mn>3<\/mn><\/msqrt><mi>cot<\/mi><mi>\u03b8<\/mi><mo>=<\/mo><mn>-1<\/mn><mo>\u27f9<\/mo><mi>cot<\/mi><mi>\u03b8<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><msqrt><mn>3<\/mn><\/msqrt><\/mfrac><\/mrow><annotation encoding=\"text\/plain\">the square root of 3 end-root cotangent theta equals negative 1 \u27f9 cotangent theta equals negative the fraction with numerator 1 and denominator the square root of 3 end-root end-fraction<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li>Reciprocal: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>tan<\/mi><mi>\u03b8<\/mi><mo>=<\/mo><mo>\u2212<\/mo><msqrt><mn>3<\/mn><\/msqrt><\/mrow><annotation encoding=\"text\/plain\">tangent theta equals negative the square root of 3 end-root<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li>Quadrant Analysis: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>tan<\/mi><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"text\/plain\">tangent theta<\/annotation><\/semantics><\/math> is negative in Quadrants II and IV.<\/li>\n\n\n\n<li>Reference Angle: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>tan<\/mi><mn>-1<\/mn><\/msup><mo>(<\/mo><msqrt><mn>3<\/mn><\/msqrt><mo>)<\/mo><mo>=<\/mo><mfrac><mi>\u03c0<\/mi><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"text\/plain\">the inverse tangent of open paren the square root of 3 end-root close paren equals the fraction with numerator pi and denominator 3 end-fraction<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li>Solutions: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><mi>\u03c0<\/mi><mo>\u2212<\/mo><mfrac><mi>\u03c0<\/mi><mn>3<\/mn><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"text\/plain\">theta equals pi minus the fraction with numerator pi and denominator 3 end-fraction equals the fraction with numerator 2 pi and denominator 3 end-fraction<\/annotation><\/semantics><\/math> (QII) and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><mo>\u2212<\/mo><mfrac><mi>\u03c0<\/mi><mn>3<\/mn><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>5<\/mn><mi>\u03c0<\/mi><\/mrow><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"text\/plain\">theta equals 2 pi minus the fraction with numerator pi and denominator 3 end-fraction equals the fraction with numerator 5 pi and denominator 3 end-fraction<\/annotation><\/semantics><\/math> (QIV).<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Problem 4: Compound Angle Application<\/strong>\n<ul class=\"wp-block-list\">\n<li><strong>Question:<\/strong> Find the exact value of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>sin<\/mi><mo>(<\/mo><msup><mn>75<\/mn><mo>\u2218<\/mo><\/msup><mo>)<\/mo><\/mrow><annotation encoding=\"text\/plain\">sine open paren 75 raised to the composed with power close paren<\/annotation><\/semantics><\/math> using compound angle formulas.<\/li>\n\n\n\n<li><strong>Solution:<\/strong>\n<ol class=\"wp-block-list\">\n<li>Use <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>sin<\/mi><mo>(<\/mo><mi>A<\/mi><mo>+<\/mo><mi>B<\/mi><mo>)<\/mo><mo>=<\/mo><mi>sin<\/mi><mi>A<\/mi><mi>cos<\/mi><mi>B<\/mi><mo>+<\/mo><mi>cos<\/mi><mi>A<\/mi><mi>sin<\/mi><mi>B<\/mi><\/mrow><annotation encoding=\"text\/plain\">sine open paren cap A plus cap B close paren equals sine cap A cosine cap B plus cosine cap A sine cap B<\/annotation><\/semantics><\/math><\/li>\n\n\n\n<li>Split <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><msup><mn>75<\/mn><mo>\u2218<\/mo><\/msup><annotation encoding=\"text\/plain\">75 raised to the composed with power<\/annotation><\/semantics><\/math>: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>sin<\/mi><mo>(<\/mo><msup><mn>45<\/mn><mo>\u2218<\/mo><\/msup><mo>+<\/mo><msup><mn>30<\/mn><mo>\u2218<\/mo><\/msup><mo>)<\/mo><\/mrow><annotation encoding=\"text\/plain\">sine open paren 45 raised to the composed with power plus 30 raised to the composed with power close paren<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li>Substitute: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>sin<\/mi><mo>(<\/mo><msup><mn>45<\/mn><mo>\u2218<\/mo><\/msup><mo>)<\/mo><mi>cos<\/mi><mo>(<\/mo><msup><mn>30<\/mn><mo>\u2218<\/mo><\/msup><mo>)<\/mo><mo>+<\/mo><mi>cos<\/mi><mo>(<\/mo><msup><mn>45<\/mn><mo>\u2218<\/mo><\/msup><mo>)<\/mo><mi>sin<\/mi><mo>(<\/mo><msup><mn>30<\/mn><mo>\u2218<\/mo><\/msup><mo>)<\/mo><\/mrow><annotation encoding=\"text\/plain\">sine open paren 45 raised to the composed with power close paren cosine open paren 30 raised to the composed with power close paren plus cosine open paren 45 raised to the composed with power close paren sine open paren 30 raised to the composed with power close paren<\/annotation><\/semantics><\/math>sin(45\u2218)cos(30\u2218)+cos(45\u2218)sin(30\u2218).<\/li>\n\n\n\n<li>Calculate: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mrow><mo>(<\/mo><mfrac><msqrt><mn>2<\/mn><\/msqrt><mn>2<\/mn><\/mfrac><mo>\u00d7<\/mo><mfrac><msqrt><mn>3<\/mn><\/msqrt><mn>2<\/mn><\/mfrac><mo>)<\/mo><\/mrow><mo>+<\/mo><mrow><mo>(<\/mo><mfrac><msqrt><mn>2<\/mn><\/msqrt><mn>2<\/mn><\/mfrac><mo>\u00d7<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo>)<\/mo><\/mrow><\/mrow><annotation encoding=\"text\/plain\">open paren the fraction with numerator the square root of 2 end-root and denominator 2 end-fraction cross the fraction with numerator the square root of 3 end-root and denominator 2 end-fraction close paren plus open paren the fraction with numerator the square root of 2 end-root and denominator 2 end-fraction cross one-half close paren<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li>Final Value: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mfrac><mrow><msqrt><mn>6<\/mn><\/msqrt><mo>+<\/mo><msqrt><mn>2<\/mn><\/msqrt><\/mrow><mn>4<\/mn><\/mfrac><annotation encoding=\"text\/plain\">the fraction with numerator the square root of 6 end-root plus the square root of 2 end-root and denominator 4 end-fraction<\/annotation><\/semantics><\/math>.\u00a0<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">********************************************************************************************************************************************************<\/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\u30e6\u30cb\u30c3\u30c82\uff08\u6570\u5b66\u7684\u624b\u6cd5\u307e\u305f\u306f\u5c02\u9580\u6570\u5b66\uff09\u306e\u300c\u4e09\u89d2\u6cd5\u3068\u95a2\u6570\u300d\u3067\u306f\u3001\u76f4\u89d2\u4e09\u89d2\u5f62\u306b\u3068\u3069\u307e\u3089\u305a\u3001\u4e09\u89d2\u95a2\u6570\u3001\u6052\u7b49\u5f0f\u3001\u30b0\u30e9\u30d5\uff08\u6b63\u5f26\u3001\u4f59\u5f26\u3001\u6b63\u63a5\u306a\u3069\uff09\u3001\u305d\u308c\u3089\u306e\u5fdc\u7528\u3001\u305d\u3057\u3066\u8907\u7d20\u6570\u3084\u30d9\u30af\u30c8\u30eb\u3092\u542b\u3080\u9ad8\u5ea6\u306a\u95a2\u6570\u306e\u6982\u5ff5\u306b\u3064\u3044\u3066\u3001\u3053\u308c\u307e\u3067\u306e\u4ee3\u6570\u3001\u30b0\u30e9\u30d5\u4f5c\u6210\u3001\u5e7e\u4f55\u5b66\u306e\u57fa\u790e\u3092\u8e0f\u307e\u3048\u3066\u7406\u89e3\u3092\u6df1\u3081\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5b66\u7fd2\u5185\u5bb9\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u95a2\u6570\uff1a\u95a2\u6570\u306f\u73fe\u5b9f\u4e16\u754c\u306e\u30b7\u30ca\u30ea\u30aa\u3092\u30e2\u30c7\u30eb\u5316\u3059\u308b\u4e0a\u3067\u4e2d\u5fc3\u7684\u306a\u5f79\u5272\u3092\u679c\u305f\u3059\u305f\u3081\u3001\u3055\u307e\u3056\u307e\u306a\u7a2e\u985e\u306e\u95a2\u6570\uff08\u7dda\u5f62\u95a2\u6570\u3001\u4e8c\u6b21\u95a2\u6570\u3001\u6307\u6570\u95a2\u6570\u306a\u3069\uff09\u3001\u95a2\u6570\u8868\u8a18\u3001\u30b0\u30e9\u30d5\u4f5c\u6210\u3001\u5909\u63db\u306b\u3064\u3044\u3066\u6df1\u304f\u6398\u308a\u4e0b\u3052\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4e09\u89d2\u6cd5\uff1a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6bd4\u3068\u6052\u7b49\u5f0f\uff1a\u57fa\u672c\u7684\u306aSOH CAH TOA\u304b\u3089\u767a\u5c55\u3057\u3001\u4e09\u89d2\u95a2\u6570\u306e\u6052\u7b49\u5f0f\uff08<math data-latex=\"sin ^{2}\\theta +\\cos ^{2}\\theta =1)\"><semantics><mrow><mi>s<\/mi><mi>i<\/mi><msup><mi>n<\/mi><mn>2<\/mn><\/msup><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><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">sin ^{2}\\theta +\\cos ^{2}\\theta =1)<\/annotation><\/semantics><\/math>\u306a\u3069\u3092\u7528\u3044\u3066\u5f0f\u3092\u7c21\u7565\u5316\u3057\u3001\u65b9\u7a0b\u5f0f\u3092\u89e3\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30b0\u30e9\u30d5\uff1a\u6b63\u5f26\u95a2\u6570\u3001\u4f59\u5f26\u95a2\u6570\u3001\u6b63\u63a5\u95a2\u6570\u306e\u5468\u671f\u6027\u3068\u30b0\u30e9\u30d5\u3092\u5206\u6790\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5fdc\u7528\uff1a\u4e09\u89d2\u6cd5\u3092\u7528\u3044\u3066\u5e7e\u4f55\u5b66\u3001\u7269\u7406\u5b66\u3001\u305d\u306e\u4ed6\u306e\u5206\u91ce\u306e\u554f\u984c\u3092\u89e3\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5c02\u9580\u6570\u5b66\u30e6\u30cb\u30c3\u30c82\u306e\u8a73\u7d30\uff08\u8a72\u5f53\u3059\u308b\u5834\u5408\uff09\uff1a\u5c02\u9580\u6570\u5b66\u30e6\u30cb\u30c3\u30c82\u3067\u306f\u3001\u95a2\u6570\u3068\u4e09\u89d2\u6cd5\u306e\u6982\u5ff5\u3092\u57fa\u76e4\u3068\u3059\u308b\u8907\u7d20\u6570\uff08\u865a\u6570\u3001\u30a2\u30eb\u30ac\u30f3\u56f3\u3001\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406\uff09\u3068\u30d9\u30af\u30c8\u30eb\u3082\u53d6\u308a\u4e0a\u3052\u308b\u5834\u5408\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u57fa\u672c\u7684\u306b\u306f\u3001\u4ee3\u6570\u95a2\u6570\u3068\u4e09\u89d2\u6cd5\u306e\u539f\u7406\u3092\u7528\u3044\u3066\u95a2\u4fc2\u3092\u8a18\u8ff0\u304a\u3088\u3073\u30e2\u30c7\u30eb\u5316\u3059\u308b\u65b9\u6cd5\u3092\u5b66\u3073\u3001\u5fae\u7a4d\u5206\u5b66\u3084\u9ad8\u5ea6\u306a\u6570\u5b66\u3078\u306e\u6e96\u5099\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">***********************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30af\u30a4\u30fc\u30f3\u30ba\u30e9\u30f3\u30c9\u5dde\u306e11\u5e74\u751f\u5c02\u9580\u6570\u5b66\u3001\u7279\u306b\u30e6\u30cb\u30c3\u30c82\u3067\u306f\u3001\u8907\u96d1\u306a\u30b0\u30e9\u30d5\u306e\u63cf\u753b\u3001\u975e\u7dda\u5f62\u4e09\u89d2\u65b9\u7a0b\u5f0f\u306e\u89e3\u3001\u6052\u7b49\u5f0f\u306e\u9069\u7528\u306a\u3069\u3001\u9ad8\u5ea6\u306a\u4e09\u89d2\u6cd5\u3068\u95a2\u6570\u306e\u6319\u52d5\u306b\u7126\u70b9\u3092\u5f53\u3066\u3066\u3044\u307e\u3059\u3002\u4e3b\u8981\u306a\u30c8\u30d4\u30c3\u30af\u306b\u306f\u3001(<math data-latex=\"\\sec\"><semantics><mrow><mi>sec<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><annotation encoding=\"application\/x-tex\">\\sec<\/annotation><\/semantics><\/math> )\u3001(\\csc )\u3001(\\cot )\u95a2\u6570\u306e\u64cd\u4f5c\u3001\u6052\u7b49\u5f0f\u306e\u8a3c\u660e\u3001\u4f4d\u76f8\u30b7\u30d5\u30c8\u3068\u5909\u63db\u3092\u7528\u3044\u305f\u30b0\u30e9\u30d5\u306e\u89e3\u6790\u306a\u3069\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30b5\u30f3\u30d7\u30eb\u554f\u984c\u3068\u89e3\u7b54<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u554f\u984c1\uff1a\u4e09\u89d2\u95a2\u6570\u306e\u30b0\u30e9\u30d5\u306e\u63cf\u753b\u3000\u554f\u984c\uff1a<math data-latex=\"(0\\le x\\le 2\\pi \"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>0<\/mn><mo>\u2264<\/mo><mi>x<\/mi><mo>\u2264<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">(0\\le x\\le 2\\pi <\/annotation><\/semantics><\/math>) \u306b\u304a\u3051\u308b (<math data-latex=\"y=3\\sin (2x-\\frac{\\pi }{2})+1\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mn>3<\/mn><mrow><mspace width=\"0.1667em\"><\/mspace><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mfrac><mi>\u03c0<\/mi><mn>2<\/mn><\/mfrac><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">y=3\\sin (2x-\\frac{\\pi }{2})+1<\/annotation><\/semantics><\/math>) \u306e\u30b0\u30e9\u30d5\u3092\u63cf\u753b\u3057\u306a\u3055\u3044\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u89e3\u7b54\uff1a\u632f\u5e45\uff1a3\u3002\u5468\u671f\uff1a(<math data-latex=\"(\\frac{2\\pi }{n}=\\frac{2\\pi }{2}=\\pi \"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mfrac><mrow><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><mi>n<\/mi><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><mn>2<\/mn><\/mfrac><mo>=<\/mo><mi>\u03c0<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">(\\frac{2\\pi }{n}=\\frac{2\\pi }{2}=\\pi <\/annotation><\/semantics><\/math> )\u3002\u4f4d\u76f8\u30b7\u30d5\u30c8\uff1a(<math data-latex=\" 2(x-\\frac{\\pi }{4})=0\\implies x=\\frac{\\pi }{4}\"><semantics><mrow><mn>2<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mfrac><mi>\u03c0<\/mi><mn>4<\/mn><\/mfrac><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>0<\/mn><mspace width=\"0.2778em\"><\/mspace><mo stretchy=\"false\">\u27f9<\/mo><mspace width=\"0.2778em\"><\/mspace><mi>x<\/mi><mo>=<\/mo><mfrac><mi>\u03c0<\/mi><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\"> 2(x-\\frac{\\pi }{4})=0\\implies x=\\frac{\\pi }{4}<\/annotation><\/semantics><\/math>)\uff08\u53f3\u306b (<math data-latex=\"\\frac{\\pi }{4}\"><semantics><mfrac><mi>\u03c0<\/mi><mn>4<\/mn><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac{\\pi }{4}<\/annotation><\/semantics><\/math>) \u30b7\u30d5\u30c8\uff09\u3002\u5782\u76f4\u30b7\u30d5\u30c8\uff1a1\u5358\u4f4d\u4e0a\uff08+1\uff09\u3002\u63cf\u753b\uff1a\u30b0\u30e9\u30d5\u306f1\u5358\u4f4d\u4e0a\u304b\u3089\u59cb\u307e\u308a\u3001\u6700\u5927\u5024\u306f (<math data-latex=\"1+3=4\"><semantics><mrow><mn>1<\/mn><mo>+<\/mo><mn>3<\/mn><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">1+3=4<\/annotation><\/semantics><\/math>)\u3001\u6700\u5c0f\u5024\u306f (<math data-latex=\"1-3=-2\"><semantics><mrow><mn>1<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">1-3=-2<\/annotation><\/semantics><\/math>) \u3067\u3059\u3002 (<math data-latex=\"x=\\frac{\\pi }{4}\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mi>\u03c0<\/mi><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x=\\frac{\\pi }{4}<\/annotation><\/semantics><\/math>) \u304b\u3089\u59cb\u307e\u308a\u3001<math data-latex=\"x=\\frac{5\\pi }{4}\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mrow><mn>5<\/mn><mi>\u03c0<\/mi><\/mrow><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x=\\frac{5\\pi }{4}<\/annotation><\/semantics><\/math> (1\u5468\u671f) \u3067\u7d42\u308f\u308b\u6b63\u5f26\u66f2\u7dda\u3092\u63cf\u304d\u3001\u7bc4\u56f2\u306b\u308f\u305f\u3063\u3066\u7e70\u308a\u8fd4\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u554f\u984c2\uff1a\u4e09\u89d2\u95a2\u6570\u306e\u6052\u7b49\u5f0f\u306e\u8a3c\u660e\u554f\u984c\uff1a(<math data-latex=\"\\frac{\\cos \\theta }{1-\\sin \\theta }-\\tan \\theta =\\sec \\theta\"><semantics><mrow><mfrac><mrow><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><\/mrow><mrow><mn>1<\/mn><mo>\u2212<\/mo><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><\/mrow><\/mfrac><mo>\u2212<\/mo><mrow><mi>tan<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><mo>=<\/mo><mrow><mi>sec<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{\\cos \\theta }{1-\\sin \\theta }-\\tan \\theta =\\sec \\theta<\/annotation><\/semantics><\/math> )\u306e\u6052\u7b49\u5f0f\u3092\u8a3c\u660e\u3057\u306a\u3055\u3044\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u89e3\u7b54\uff1a\u5de6\u8fba\u304b\u3089\u59cb\u3081\u308b\uff1a(<math data-latex=\"\\frac{\\cos \\theta }{1-\\sin \\theta }-\\frac{\\sin \\theta }{\\cos \\theta }\"><semantics><mrow><mfrac><mrow><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><\/mrow><mrow><mn>1<\/mn><mo>\u2212<\/mo><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><\/mrow><\/mfrac><mo>\u2212<\/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\">\\frac{\\cos \\theta }{1-\\sin \\theta }-\\frac{\\sin \\theta }{\\cos \\theta }<\/annotation><\/semantics><\/math>)\u3002\u516c\u5206\u6bcd\uff1a(<math data-latex=\"\\frac{\\cos ^{2}\\theta -\\sin \\theta (1-\\sin \\theta )}{\\cos \\theta (1-\\sin \\theta )}\"><semantics><mfrac><mrow><msup><mi>cos<\/mi><mn>2<\/mn><\/msup><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><mi>\u03b8<\/mi><mo>\u2212<\/mo><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo><\/mrow><mrow><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo><\/mrow><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac{\\cos ^{2}\\theta -\\sin \\theta (1-\\sin \\theta )}{\\cos \\theta (1-\\sin \\theta )}<\/annotation><\/semantics><\/math>)\u3002\u5c55\u958b\uff1a(<math data-latex=\"\\frac{\\cos ^{2}\\theta -\\sin \\theta +\\sin ^{2}\\theta }{\\cos \\theta (1-\\sin \\theta )}\"><semantics><mfrac><mrow><msup><mi>cos<\/mi><mn>2<\/mn><\/msup><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><mi>\u03b8<\/mi><mo>\u2212<\/mo><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><mo>+<\/mo><msup><mi>sin<\/mi><mn>2<\/mn><\/msup><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><mi>\u03b8<\/mi><\/mrow><mrow><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo><\/mrow><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac{\\cos ^{2}\\theta -\\sin \\theta +\\sin ^{2}\\theta }{\\cos \\theta (1-\\sin \\theta )}<\/annotation><\/semantics><\/math>). (<math data-latex=\"\\cos ^{2}\\theta +\\sin ^{2}\\theta =1\"><semantics><mrow><msup><mi>cos<\/mi><mn>2<\/mn><\/msup><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><mi>\u03b8<\/mi><mo>+<\/mo><msup><mi>sin<\/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\">\\cos ^{2}\\theta +\\sin ^{2}\\theta =1<\/annotation><\/semantics><\/math>)\u3092\u4f7f\u3063\u3066: (<math data-latex=\"\\frac{1-\\sin \\theta }{\\cos \\theta (1-\\sin \\theta )}\"><semantics><mfrac><mrow><mn>1<\/mn><mo>\u2212<\/mo><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><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo><\/mrow><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac{1-\\sin \\theta }{\\cos \\theta (1-\\sin \\theta )}<\/annotation><\/semantics><\/math>). \u30ad\u30e3\u30f3\u30bb\u30eb (<math data-latex=\"1-\\sin \\theta \"><semantics><mrow><mn>1<\/mn><mo>\u2212<\/mo><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">1-\\sin \\theta <\/annotation><\/semantics><\/math>): (<math data-latex=\"\\frac{1}{\\cos \\theta }=\\sec \\theta =\\text{RHS}\"><semantics><mrow><mfrac><mn>1<\/mn><mrow><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mrow><mi>sec<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><mo>=<\/mo><mtext>RHS<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{1}{\\cos \\theta }=\\sec \\theta =\\text{RHS}<\/annotation><\/semantics><\/math>).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u554f\u984c3\uff1a\u4e09\u89d2\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5\u554f\u984c\uff1a(<math data-latex=\"\\sqrt{3}\\cot \\theta +1=0\"><semantics><mrow><msqrt><mn>3<\/mn><\/msqrt><mrow><mi>cot<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt{3}\\cot \\theta +1=0<\/annotation><\/semantics><\/math>)\u3092(<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> )\u306b\u3064\u3044\u3066\u89e3\u304d\u306a\u3055\u3044\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u89e3\u7b54\uff1a\u5909\u5f62\uff1a(<math data-latex=\"\\sqrt{3}\\cot \\theta =-1\\implies \\cot \\theta =-\\frac{1}{\\sqrt{3}}\"><semantics><mrow><msqrt><mn>3<\/mn><\/msqrt><mrow><mi>cot<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mspace width=\"0.2778em\"><\/mspace><mo stretchy=\"false\">\u27f9<\/mo><mspace width=\"0.2778em\"><\/mspace><mrow><mspace width=\"0.1667em\"><\/mspace><mi>cot<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mfrac><mn>1<\/mn><msqrt><mn>3<\/mn><\/msqrt><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt{3}\\cot \\theta =-1\\implies \\cot \\theta =-\\frac{1}{\\sqrt{3}}<\/annotation><\/semantics><\/math>)\u3002\u9006\u6570\uff1a(<math data-latex=\"\\tan \\theta =-\\sqrt{3}\"><semantics><mrow><mrow><mi>tan<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><msqrt><mn>3<\/mn><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\tan \\theta =-\\sqrt{3}<\/annotation><\/semantics><\/math>)\u3002\u8c61\u9650\u5206\u6790\uff1a(<math data-latex=\"\\tan \\theta \"><semantics><mrow><mrow><mi>tan<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\tan \\theta <\/annotation><\/semantics><\/math>)\u306f\u7b2cII\u8c61\u9650\u3068\u7b2cIV\u8c61\u9650\u3067\u306f\u8ca0\u306e\u5024\u3068\u306a\u308b\u3002\u57fa\u6e96\u89d2\uff1a<math data-latex=\" \\tan ^{-1}(\\sqrt{3})=\\frac{\\pi }{3}\"><semantics><mrow><msup><mi>tan<\/mi><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo>\u2061<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msqrt><mn>3<\/mn><\/msqrt><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mi>\u03c0<\/mi><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\"> \\tan ^{-1}(\\sqrt{3})=\\frac{\\pi }{3}<\/annotation><\/semantics><\/math>\u3002\u89e3\u7b54\uff1a(<math data-latex=\"\\theta =\\pi -\\frac{\\pi }{3}=\\frac{2\\pi }{3})\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><mi>\u03c0<\/mi><mo>\u2212<\/mo><mfrac><mi>\u03c0<\/mi><mn>3<\/mn><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><mn>3<\/mn><\/mfrac><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\theta =\\pi -\\frac{\\pi }{3}=\\frac{2\\pi }{3})<\/annotation><\/semantics><\/math> (QII) \u304a\u3088\u3073 (<math data-latex=\"\\theta =2\\pi -\\frac{\\pi }{3}=\\frac{5\\pi }{3}\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><mo>\u2212<\/mo><mfrac><mi>\u03c0<\/mi><mn>3<\/mn><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>5<\/mn><mi>\u03c0<\/mi><\/mrow><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\theta =2\\pi -\\frac{\\pi }{3}=\\frac{5\\pi }{3}<\/annotation><\/semantics><\/math>) (QIV)\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u554f\u984c4\uff1a\u8907\u5408\u89d2\u306e\u5fdc\u7528\u554f\u984c\uff1a\u8907\u5408\u89d2\u306e\u516c\u5f0f\u3092\u4f7f\u7528\u3057\u3066\u3001(<math data-latex=\"\\sin (75^{\\circ })\"><semantics><mrow><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mn>75<\/mn><mo lspace=\"0em\" rspace=\"0em\">\u2218<\/mo><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\sin (75^{\\circ })<\/annotation><\/semantics><\/math>)\u306e\u6b63\u78ba\u306a\u5024\u3092\u6c42\u3081\u307e\u3059\u3002\u89e3\u7b54\uff1a(<math data-latex=\"\\sin (A+B)=\\sin A\\cos B+\\cos A\\sin B\"><semantics><mrow><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>A<\/mi><mo>+<\/mo><mi>B<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>A<\/mi><mrow><mspace width=\"0.1667em\"><\/mspace><mi>cos<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>B<\/mi><mo>+<\/mo><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>A<\/mi><mrow><mspace width=\"0.1667em\"><\/mspace><mi>sin<\/mi><mo>\u2061<\/mo><mspace width=\"0.1667em\"><\/mspace><\/mrow><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\sin (A+B)=\\sin A\\cos B+\\cos A\\sin B<\/annotation><\/semantics><\/math>)\u3092\u4f7f\u7528\u3057\u307e\u3059\u3002    (<math data-latex=\"75^{\\circ }\"><semantics><msup><mn>75<\/mn><mo lspace=\"0em\" rspace=\"0em\">\u2218<\/mo><\/msup><annotation encoding=\"application\/x-tex\">75^{\\circ }<\/annotation><\/semantics><\/math>)\u3092\u5206\u5272\u3057\u307e\u3059\uff1a(<math data-latex=\"\\sin (45^{\\circ }+30^{\\circ })\"><semantics><mrow><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mn>45<\/mn><mo lspace=\"0em\" rspace=\"0em\">\u2218<\/mo><\/msup><mo>+<\/mo><msup><mn>30<\/mn><mo lspace=\"0em\" rspace=\"0em\">\u2218<\/mo><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\sin (45^{\\circ }+30^{\\circ })<\/annotation><\/semantics><\/math>)\u3002\u4ee3\u5165\uff1a<math data-latex=\"\\sin (45^{\\circ })\\cos (30^{\\circ })+\\cos (45^{\\circ })\\sin (30^{\\circ })\"><semantics><mrow><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mn>45<\/mn><mo lspace=\"0em\" rspace=\"0em\">\u2218<\/mo><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mrow><mspace width=\"0.1667em\"><\/mspace><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mn>30<\/mn><mo lspace=\"0em\" rspace=\"0em\">\u2218<\/mo><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mrow><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mn>45<\/mn><mo lspace=\"0em\" rspace=\"0em\">\u2218<\/mo><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mrow><mspace width=\"0.1667em\"><\/mspace><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mn>30<\/mn><mo lspace=\"0em\" rspace=\"0em\">\u2218<\/mo><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\sin (45^{\\circ })\\cos (30^{\\circ })+\\cos (45^{\\circ })\\sin (30^{\\circ })<\/annotation><\/semantics><\/math>\u3002\u8a08\u7b97\uff1a(<math data-latex=\"\\left(\\frac{\\sqrt{2}}{2}\\times \\frac{\\sqrt{3}}{2}\\right)+\\left(\\frac{\\sqrt{2}}{2}\\times \\frac{1}{2}\\right)\"><semantics><mrow><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><mfrac><msqrt><mn>2<\/mn><\/msqrt><mn>2<\/mn><\/mfrac><mo>\u00d7<\/mo><mfrac><msqrt><mn>3<\/mn><\/msqrt><mn>2<\/mn><\/mfrac><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo>+<\/mo><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><mfrac><msqrt><mn>2<\/mn><\/msqrt><mn>2<\/mn><\/mfrac><mo>\u00d7<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\left(\\frac{\\sqrt{2}}{2}\\times \\frac{\\sqrt{3}}{2}\\right)+\\left(\\frac{\\sqrt{2}}{2}\\times \\frac{1}{2}\\right)<\/annotation><\/semantics><\/math>).\u6700\u7d42\u5024: (<math data-latex=\"\\frac{\\sqrt{6}+\\sqrt{2}}{4}\"><semantics><mfrac><mrow><msqrt><mn>6<\/mn><\/msqrt><mo>+<\/mo><msqrt><mn>2<\/mn><\/msqrt><\/mrow><mn>4<\/mn><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac{\\sqrt{6}+\\sqrt{2}}{4}<\/annotation><\/semantics><\/math>).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In Queensland Grade 11 Mathematics, &#8220;Trigonometry and Functions&#8221; in Unit 2 (likely for Mathematical Methods or Specialist Mathematics) involves extending beyond right-angled triangles to understand trigonometric functions, identities, graphs (like sine, cosine, tangent), their applications, and advanced function concepts, including potentially complex numbers and vectors, building on earlier algebra, graphing, and geometric foundations.&nbsp; Here&#8217;s a [&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-1206","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1206","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=1206"}],"version-history":[{"count":5,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1206\/revisions"}],"predecessor-version":[{"id":1244,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1206\/revisions\/1244"}],"wp:attachment":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1206"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1206"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1206"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}