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 5.6

Textbook Question

Match each expression in Column I with its equivalent expression in Column II.
sin 60° cos 45° - cos 60° sin 45°

Verified step by step guidance

Recognize that the expression \( \sin 60^\circ \cos 45^\circ - \cos 60^\circ \sin 45^\circ \) resembles the sine difference identity.

Recall the sine difference identity: \( \sin(A - B) = \sin A \cos B - \cos A \sin B \).

Identify \( A = 60^\circ \) and \( B = 45^\circ \) in the expression.

Substitute \( A \) and \( B \) into the sine difference identity: \( \sin(60^\circ - 45^\circ) \).

Simplify the expression to find the equivalent expression in Column II.

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Key Concepts

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

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. One important identity is the sine difference identity, which states that sin(A - B) = sin(A)cos(B) - cos(A)sin(B). This identity is crucial for simplifying expressions involving sine and cosine functions.

Angle Measures in Degrees

In trigonometry, angles are often measured in degrees, where a full circle is 360 degrees. Specific angles, such as 30°, 45°, and 60°, have known sine and cosine values that are frequently used in calculations. Understanding these values is essential for evaluating trigonometric expressions and solving problems.

Sine and Cosine Functions

The sine and cosine functions are fundamental in trigonometry, representing the ratios of the sides of a right triangle relative to its angles. For any angle θ, sin(θ) is the ratio of the opposite side to the hypotenuse, while cos(θ) is the ratio of the adjacent side to the hypotenuse. These functions are periodic and have specific values for common angles, which are vital for solving trigonometric equations.