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

Introduction to Trigonometric Identities

Trigonometry

6. Trigonometric Identities and More Equations

Introduction to Trigonometric Identities

Previous problemNext problem

1:52 minutes

Problem 9

Textbook Question

Be sure that you've familiarized yourself with the second set of formulas presented in this section by working C5–C8 in the Concept and Vocabulary Check. In Exercises 9–22, express each sum or difference as a product. If possible, find this product's exact value.sin 6x + sin 2x

Verified step by step guidance

Identify the sum-to-product identity for sine: \( \sin A + \sin B = 2 \sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \).

Assign \( A = 6x \) and \( B = 2x \) to the identity.

Calculate \( \frac{A + B}{2} = \frac{6x + 2x}{2} = 4x \).

Calculate \( \frac{A - B}{2} = \frac{6x - 2x}{2} = 2x \).

Express the sum as a product: \( 2 \sin(4x) \cos(2x) \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

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

Sum-to-Product Formulas

Sum-to-product formulas are trigonometric identities that express sums or differences of sine and cosine functions as products. For example, the formula for the sum of two sine functions is sin A + sin B = 2 sin((A + B)/2) cos((A - B)/2). These formulas simplify the process of solving trigonometric equations and are essential for transforming expressions like sin 6x + sin 2x into a more manageable form.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions refer to the specific values of sine, cosine, and tangent at key angles, such as 0°, 30°, 45°, 60°, and 90°. Knowing these values allows for the simplification of trigonometric expressions and calculations. In the context of the given problem, finding the exact value of the product resulting from the sum-to-product transformation is crucial for a complete solution.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variables involved, providing relationships between different trigonometric functions. Familiarity with these identities, such as the Pythagorean identities, angle sum and difference identities, and double angle formulas, is vital for manipulating and simplifying trigonometric expressions. They serve as foundational tools for solving problems in trigonometry, including the one presented.