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 64b

Chapter 3, Problem 64b

In Exercises 57–64, find the exact value of the following under the given conditions: b. sin (α + β), sin α = 5/6 , 𝝅/2 < α < 𝝅 , and tan β = 3/7 , 𝝅 < β < 3𝝅/2 .

Verified step by step guidance

Identify the given information: $ \sin \alpha = \frac{5\pi}{6} $ with $ \frac{3\pi}{2} < \alpha < \pi $, and $ \tan \beta = \frac{3\pi}{2} $ with $ \frac{7\pi}{2} < \beta < \text{(missing upper bound)} $. Note that the interval for $ \alpha $ seems inconsistent since $ \frac{3\pi}{2} > \pi $. Verify the intervals and values carefully before proceeding.

Recall the formula for $ \sin(\alpha + \beta) $: \[ \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \]

Since $ \sin \alpha $ is given, find $ \cos \alpha $ using the Pythagorean identity: \[ \cos \alpha = \pm \sqrt{1 - \sin^2 \alpha} \] Determine the correct sign of $ \cos \alpha $ based on the quadrant where $ \alpha $ lies.

Use the given $ \tan \beta $ to find $ \sin \beta $ and $ \cos \beta $. Recall that: \[ \tan \beta = \frac{\sin \beta}{\cos \beta} \] Use the Pythagorean identity to express $ \sin \beta $ and $ \cos \beta $ in terms of $ \tan \beta $, and determine their signs based on the quadrant of $ \beta $.

Substitute the values of $ \sin \alpha $, $ \cos \alpha $, $ \sin \beta $, and $ \cos \beta $ into the formula for $ \sin(\alpha + \beta) $ and simplify to find the exact 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.

Sum of Angles Formula for Sine

The sum of angles formula states that sin(α + β) = sin α cos β + cos α sin β. This identity allows us to find the sine of a sum of two angles using the sines and cosines of the individual angles, which is essential when given trigonometric values of α and β separately.

Determining the Sign of Trigonometric Functions Based on Quadrants

The signs of sine, cosine, and tangent depend on the quadrant in which the angle lies. Knowing the interval for α and β helps determine whether sine, cosine, or tangent values are positive or negative, which is crucial for correctly evaluating trigonometric expressions.

Using Given Trigonometric Ratios to Find Missing Values

Given sin α and tan β, we can use Pythagorean identities to find cos α and cos β. For example, cos α = ±√(1 - sin² α), with the sign determined by the quadrant. Similarly, from tan β, we find sin β and cos β using the identity tan β = sin β / cos β, enabling full evaluation of sin(α + β).

Recommended video:

5:13

Finding Missing Angles

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. 5 cos² x - 3 = 0

views

Textbook Question

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 25–32, write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. sin 40° cos 20° + cos 40° sin 20°

Textbook Question

In Exercises 59–68, verify each identity. $\cot \frac{x}{2} = \frac{\sin x}{1 - \cos x}$

views

Textbook Question

In Exercises 57–64, find the exact value of the following under the given conditions: c. tan (α + β), sin α = 5/6 , 𝝅/2 < α < 𝝅 , and tan β = 3/7 , 𝝅 < β < 3𝝅/2 .

Textbook Question

In Exercises 39–46, use a half-angle formula to find the exact value of each expression. cos 22.5°

views

Textbook Question

Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). cos 4x = ﹣√3 / 2

views