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 3.RE.35b

Chapter 3, Problem 3.RE.35b

In Exercises 35–38, find the exact value of the following under the given conditions: b. cos(α﹣β)
sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.

Verified step by step guidance

Identify the given information: \( \sin \alpha = \frac{3}{5} \) with \( 0 < \alpha < \frac{\pi}{2} \), and \( \sin \beta = \frac{12}{13} \) with \( \frac{\pi}{2} < \beta < \pi \).

Determine the quadrant of each angle to find the signs of \( \cos \alpha \) and \( \cos \beta \). Since \( \alpha \) is in the first quadrant, \( \cos \alpha > 0 \). Since \( \beta \) is in the second quadrant, \( \cos \beta < 0 \).

Use the Pythagorean identity to find \( \cos \alpha \) and \( \cos \beta \): \[ \cos \theta = \pm \sqrt{1 - \sin^2 \theta} \] Calculate: \[ \cos \alpha = +\sqrt{1 - \left(\frac{3}{5}\right)^2} \] \[ \cos \beta = -\sqrt{1 - \left(\frac{12}{13}\right)^2} \]

Recall the cosine difference identity: \[ \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \]

Substitute the values of \( \sin \alpha \), \( \sin \beta \), \( \cos \alpha \), and \( \cos \beta \) into the identity to express \( \cos(\alpha - \beta) \) exactly.

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.

Sum and Difference Formulas for Cosine

The cosine of the difference of two angles, cos(α - β), can be found using the formula cos(α - β) = cos α cos β + sin α sin β. This identity allows us to express the cosine of a difference in terms of the sines and cosines of the individual angles, which is essential when given sine values and angle ranges.

Determining Cosine from Sine and Quadrant Information

Given sin α and sin β along with their angle ranges, we can find cos α and cos β using the Pythagorean identity cos²θ = 1 - sin²θ. The sign of cosine depends on the quadrant of the angle, so knowing the interval for α and β helps determine whether cosine is positive or negative.

Angle Measurement in Radians and Quadrant Boundaries

Angles are given in radians with specified intervals (e.g., 0 < α < 3π/2). Understanding these intervals helps identify the quadrant in which each angle lies, which is crucial for determining the signs of trigonometric functions and correctly applying identities.

Recommended video:

5:04

Converting between Degrees & Radians

Related Practice

Textbook Question

In Exercises 54–67, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. sin 2x = √ 3 sin x

658

views

Textbook Question

In Exercises 39–42, use double- and half-angle formulas to find the exact value of each expression. sin 22.5°

729

views

Textbook Question

In Exercises 35–38, find the exact value of the following under the given conditions:

b. cos(α﹣β)

sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2.

992

views

Textbook Question

In Exercises 54–67, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. cos 2x = -1

527

views

Textbook Question

In Exercises 35–38, find the exact value of the following under the given conditions:

c. tan(α + β)

sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.

843

views

Textbook Question

In Exercises 50–53, find all solutions of each equation. cos x = ﹣1/2

485

views