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 13

Chapter 3, Problem 13

Solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. sin 2x + cos x = 0

Verified step by step guidance

Start by rewriting the given equation: \(\sin 2x + \cos x = 0\).

Use the double-angle identity for sine: \(\sin 2x = 2 \sin x \cos x\). Substitute this into the equation to get \(2 \sin x \cos x + \cos x = 0\).

Factor out the common term \(\cos x\): \(\cos x (2 \sin x + 1) = 0\).

Set each factor equal to zero and solve separately: 1) \(\cos x = 0\) 2) \(2 \sin x + 1 = 0\).

Solve each equation on the interval \([0, 2\pi)\): - For \(\cos x = 0\), find all \(x\) where cosine is zero. - For \(2 \sin x + 1 = 0\), isolate \(\sin x\) and find all \(x\) where sine equals that value.

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 Identity for Sine

The double-angle identity expresses sin(2x) as 2 sin(x) cos(x). This allows rewriting the equation sin 2x + cos x = 0 in terms of sin(x) and cos(x), simplifying the solving process by reducing it to a single trigonometric function or a product of functions.

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle solutions within the given interval. This often requires factoring, using identities, and considering the periodicity of sine and cosine to find all valid solutions between 0 and 2π.

Interval Restriction and Exact Values

The problem restricts solutions to the interval [0, 2π), meaning only angles within one full rotation are considered. Solutions should be given as exact values (like π/3) or approximated to four decimal places, ensuring clarity and precision in the final answers.

Recommended video:

5:10

Evaluate Composite Functions - Values on Unit Circle

Related Practice

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(60° - 45°)

views

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. cos 4x + cos 2x

views

Textbook Question

Use the given information to find the exact value of each of the following: tan 2θ

cot θ = 2, θ lies in quadrant III.

views

Textbook Question

In Exercises 7–14, use the given information to find the exact value of each of the following:c. tan 2θcot θ = 3, θ lies in quadrant III.

Textbook Question

Use the given information to find the exact value of each of the following: cos 2θ

cot θ = 2, θ lies in quadrant III.

Textbook Question

Find all solutions of each equation. tan x = 1

views