Trigonometry - Trigonometric Identities
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This is a topic under Grade 11 Pre-Calculus. This will discuss proving and simplifying Trigonometric Identities.
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The document provides information about trigonometry concepts including definitions of sine, cosine, tangent, cotangent, secant and cosecant using right triangles. It discusses key formulas for trigonometric identities, addition and subtraction of angles, multiplication of angles, and trigonometric ratios. Examples of problems and their step-by-step solutions are also presented.
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This document provides an overview of trigonometric functions and identities involving reciprocal trig functions such as secant, cosecant, and cotangent. It defines these reciprocal functions, shows how they relate to the original trig functions of sine, cosine, and tangent, and sketches their graphs. It also presents new trigonometric identities involving secant, cosecant, and cotangent, provides examples of calculations and proofs involving these functions, and gives some example problems to solve.
角度と三角比（人間科学のための基礎数学）
過去に某大学某学部の1回生向け授業のために作ったスライドを改変したものです。文系出身で数学IAしか習っていない学生も少なからずいたため、高校数学の範囲の説明を含んでいます（親切な説明ではないかもしれませんが）
実際の授業では三角関数とセットにしていたため、三角比単品での実用編は抑えめになり、今回分では動作解析での関節角度だけになりました。実際に3次元の動作解析する場合は余弦定理だけでは済まなくなることもありますが、ここでは肘や膝のような蝶番関節の角度ということでご容赦下さい…
Tugas 5.3 kalkulus integral
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2. Exercises 55-62 involve sketching graphs of functions and finding their average values over given intervals.
3. Exercises 63-70 involve using the integral definition as a limit of Riemann sums to evaluate definite integrals.
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- 2. Classification of Equations • Conditional Equation – only true for some values of the variable. Ex. cos 𝜃 = sin 𝜃 is true for 𝜃 = 45° but not for 𝜃 = 30° • Identity – true for all values of the variable. Ex. tan 𝜃 cos 𝜃 = sin 𝜃
- 4. EXAMPLES: PROVE THE FOLLOWING 2. 𝑡𝑎𝑛𝜃 𝑠𝑖𝑛𝜃 = 𝑠𝑒𝑐𝜃 3. 𝑐𝑠𝑐𝜃sin𝜃 = cos𝜃𝑠𝑒𝑐𝜃 4. 𝑠𝑖𝑛𝜃 𝑡𝑎𝑛𝜃 1 𝑠𝑒𝑐𝜃 = 1 − 𝑠𝑖𝑛2 𝜃 5. 𝑡𝑎𝑛2 𝑥 = 𝑠𝑖𝑛2 𝑥𝑐𝑜𝑠−2 𝑥 6. 1−𝑐𝑜𝑠2 𝜃 𝑠𝑖𝑛2 𝜃 = 𝑐𝑜𝑠2 𝜃 + 𝑠𝑖𝑛2 𝜃 7. 𝑐𝑠𝑐2 𝜃 + 𝑠𝑒𝑐2 𝜃 = (1 − 𝑐𝑜𝑠2 𝜃) −1 (1 − 𝑠𝑖𝑛2 𝜃) −1
- 5. PROVE THE FOLLOWING EQUATIONS
- 6. 1. 𝑐𝑜𝑠2 𝑥 − 𝑠𝑖𝑛2 𝑥 = 𝑐𝑜𝑠4 𝑥 − 𝑠𝑖𝑛4 𝑥 2. 𝑡𝑎𝑛𝑥 + 1 𝑡𝑎𝑛𝑥 = 1 𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑥 3. 𝑠𝑖𝑛2 𝑥 1−𝑐𝑜𝑠𝑥 = 1 + 𝑐𝑜𝑠𝑥 4. tan 𝜃 sin 𝜃 cot 𝜃 cos 𝜃 = 𝑡𝑎𝑛3 𝜃
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- 8. 1. sec 𝑥−cos 𝑥 sec 𝑥 2. (tan 𝑥 + cot 𝑥) sin x cos x 3. 1+csc 𝑥 cos 𝑥+cot 𝑥 4. 𝑐𝑜𝑠2 𝑥 1−sin 𝑥 5. (1−𝑐𝑜𝑠2 𝑥)2 𝑠𝑖𝑛2 𝑥+𝑠𝑖𝑛2 𝑥𝑐𝑜𝑠𝑥