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 5.18

Textbook Question

Find the exact value of each expression. (Do not use a calculator.)
cos (7π/9) cos (2π/9) - sin (7π/9) sin (2π/9)

Verified step by step guidance

Recognize the expression cos(A)cos(B) - sin(A)sin(B) as a trigonometric identity.

Recall the cosine addition formula: \( \cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B) \).

Identify A as \( \frac{7\pi}{9} \) and B as \( \frac{2\pi}{9} \).

Substitute A and B into the cosine addition formula: \( \cos\left(\frac{7\pi}{9} + \frac{2\pi}{9}\right) \).

Simplify the expression inside the cosine: \( \cos\left(\frac{9\pi}{9}\right) = \cos(\pi) \).

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Key Concepts

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

Cosine and Sine Addition Formulas

The cosine and sine addition formulas are fundamental identities in trigonometry that express the cosine and sine of the sum of two angles. Specifically, cos(a + b) = cos(a)cos(b) - sin(a)sin(b) and sin(a + b) = sin(a)cos(b) + cos(a)sin(b). These formulas allow us to simplify expressions involving trigonometric functions of sums of angles, which is essential for solving the given problem.

Reference Angles

Reference angles are the acute angles formed by the terminal side of an angle in standard position and the x-axis. They help in determining the values of trigonometric functions for angles that are not standard. In this problem, understanding the reference angles for 7π/9 and 2π/9 is crucial for evaluating the cosine and sine functions accurately.

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is a key tool in trigonometry for defining the sine and cosine of angles. By using the unit circle, we can find the exact values of trigonometric functions for various angles, including those expressed in radians like 7π/9 and 2π/9, which are necessary for solving the expression in the question.