Table of contents
- 0. Review of College Algebra4h 43m
- 1. Measuring Angles39m
- 2. Trigonometric Functions on Right Triangles2h 5m
- 3. Unit Circle1h 19m
- 4. Graphing Trigonometric Functions1h 19m
- 5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
- 6. Trigonometric Identities and More Equations2h 34m
- 7. Non-Right Triangles1h 38m
- 8. Vectors2h 25m
- 9. Polar Equations2h 5m
- 10. Parametric Equations1h 6m
- 11. Graphing Complex Numbers1h 7m
6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Problem 5.16c
Textbook Question
Textbook QuestionUse a half-angle identity to find each exact value.
sin 165°
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Half-Angle Identities
Half-angle identities are trigonometric formulas that express the sine, cosine, and tangent of half an angle in terms of the trigonometric functions of the original angle. For sine, the identity is sin(θ/2) = ±√((1 - cos(θ))/2). These identities are particularly useful for finding exact values of trigonometric functions at angles that are not standard, such as 165°.
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Reference Angles
A reference angle is the acute angle formed by the terminal side of an angle and the x-axis. For angles greater than 90° but less than 360°, the reference angle helps in determining the sine, cosine, and tangent values by relating them to the corresponding acute angle. For sin(165°), the reference angle is 15°, which is essential for applying the half-angle identity.
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Exact Values of Trigonometric Functions
Exact values of trigonometric functions refer to the precise values of sine, cosine, and tangent for specific angles, often expressed in terms of square roots. For example, sin(30°) = 1/2 and cos(45°) = √2/2. Knowing these exact values allows for the simplification of expressions and calculations when using identities, such as when calculating sin(165°) using the half-angle identity.
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