Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
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

Sum and Difference Identities

Trigonometry

6. Trigonometric Identities and More Equations

Sum and Difference Identities

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Problem 5.20

Textbook Question

Write each function value in terms of the cofunction of a complementary angle.
sin 15°

Verified step by step guidance

Recall the cofunction identity: \( \sin(\theta) = \cos(90^\circ - \theta) \).

Identify the given angle \( \theta = 15^\circ \).

Substitute \( \theta = 15^\circ \) into the cofunction identity: \( \sin(15^\circ) = \cos(90^\circ - 15^\circ) \).

Simplify the expression inside the cosine function: \( 90^\circ - 15^\circ = 75^\circ \).

Express \( \sin(15^\circ) \) in terms of the cofunction: \( \sin(15^\circ) = \cos(75^\circ) \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Cofunction Identities

Cofunction identities relate the trigonometric functions of complementary angles. For example, the sine of an angle is equal to the cosine of its complement: sin(θ) = cos(90° - θ). This relationship is crucial for expressing trigonometric functions in terms of their cofunctions, particularly when dealing with angles that sum to 90 degrees.

Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees. In trigonometry, understanding complementary angles is essential for applying cofunction identities. For instance, if you have an angle of 15°, its complement is 75°, which can be used to express trigonometric functions in a different form.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are fundamental in trigonometry, representing the ratios of sides in a right triangle. Each function has specific properties and relationships, including the ability to express one function in terms of another using identities. This is particularly useful when simplifying expressions or solving equations involving angles.