Ch. 3 - Trigonometric Identities and Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 3 - Trigonometric Identities and Equations

Problem 15

Chapter 3, Problem 15

In Exercises 15–22, write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. 2 sin 15° cos 15°

Verified step by step guidance

Recognize that the expression \(2 \sin 15^\circ \cos 15^\circ\) matches the double-angle identity for sine, which is \(2 \sin A \cos A = \sin 2A\).

Rewrite the expression using the identity: \(2 \sin 15^\circ \cos 15^\circ = \sin (2 \times 15^\circ)\).

Simplify the angle inside the sine function: \(\sin (30^\circ)\).

Recall the exact value of \(\sin 30^\circ\), which is a commonly known special angle in trigonometry.

Conclude that the exact value of the original expression is the exact value of \(\sin 30^\circ\).

Verified video answer for a similar problem:

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

Video duration:

Was this helpful?

Key Concepts

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

Double-Angle Identities

Double-angle identities express trigonometric functions of twice an angle in terms of functions of the original angle. For sine, the identity is sin(2θ) = 2 sin θ cos θ, which allows rewriting products like 2 sin 15° cos 15° as sin 30°.

Evaluating Exact Trigonometric Values

Exact values of trigonometric functions at special angles (like 30°, 45°, 60°) are well-known and can be used to find precise results without a calculator. For example, sin 30° equals 1/2, which helps in determining the exact value of the expression.

Trigonometric Function Notation and Simplification

Understanding how to rewrite expressions using trigonometric notation and simplify them is essential. Recognizing patterns such as products of sine and cosine that match double-angle formulas enables efficient simplification and evaluation.

Recommended video:

06:01

i & j Notation

Related Practice

Textbook Question

In Exercises 14–19, use a sum or difference formula to find the exact value of each expression. sin 195°

Textbook Question

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. sin 75°

views

Textbook Question

Find all solutions of each equation. tan x = 0

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 x + sin 2x

views

Textbook Question

Verify each identity. cos² θ (1 + tan² θ) = 1

views

Textbook Question

In Exercises 12–18, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. 2 sin² x + cos x = 1