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Ch. 5 - Trigonometric Identities
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 5 - Trigonometric IdentitiesProblem 5.RE.30a
Chapter 6, Problem 5.RE.30a

Use the given information to find sin(x + y).
sin y = - 2/3 , cos x = - 1/5, x in quadrant II, y in quadrant III

Verified step by step guidance
1
Identify the given information: \(\sin y = -\frac{2}{3}\), \(\cos x = -\frac{1}{5}\), with \(x\) in quadrant II and \(y\) in quadrant III.
Use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to find \(\cos y\). Since \(\sin y = -\frac{2}{3}\), calculate \(\cos y = \pm \sqrt{1 - \sin^2 y} = \pm \sqrt{1 - \left(-\frac{2}{3}\right)^2}\), and determine the correct sign based on the quadrant of \(y\).
Similarly, find \(\sin x\) using \(\cos x = -\frac{1}{5}\). Calculate \(\sin x = \pm \sqrt{1 - \cos^2 x} = \pm \sqrt{1 - \left(-\frac{1}{5}\right)^2}\), and choose the sign according to the quadrant of \(x\).
Recall the angle addition formula for sine: \(\sin(x + y) = \sin x \cos y + \cos x \sin y\).
Substitute the values of \(\sin x\), \(\cos y\), \(\cos x\), and \(\sin y\) into the formula to express \(\sin(x + y)\) in terms of known quantities.

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Key Concepts

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

Sum of Angles Formula for Sine

The sine of the sum of two angles, sin(x + y), can be found using the formula sin(x + y) = sin x cos y + cos x sin y. This identity allows us to express sin(x + y) in terms of the sines and cosines of the individual angles x and y.
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Verifying Identities with Sum and Difference Formulas

Determining Signs of Trigonometric Functions by Quadrant

The signs of sine and cosine depend on the quadrant in which the angle lies. In quadrant II, sine is positive and cosine is negative; in quadrant III, both sine and cosine are negative. This helps determine the correct values and signs of unknown trigonometric functions.
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Introduction to Trigonometric Functions

Using Pythagorean Identity to Find Missing Values

The Pythagorean identity, sin²θ + cos²θ = 1, allows us to find an unknown sine or cosine value when the other is given. By substituting the known value and solving, we can find the missing trigonometric function needed to apply the sum formula.
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Pythagorean Identities
Related Practice
Textbook Question

Verify that each equation is an identity.

(sin⁴ α - cos⁴ α )/(sin² α - cos² α) = 1

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Textbook Question

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.

csc x - cot x

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Textbook Question

For each expression in Column I, choose the expression from Column II that completes an identity.

6. sec² x = ____


II

A. sin ^2 x/cos ^2 x

B.1/(sec ^2 x)

C. sin (-x)

D. csc ^2 x-cot ^2 x + sin ^2 x

E. tan x

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Textbook Question

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.

(cos x sin 2x)/1 + cos 2x)

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Textbook Question

Find values of the sine and cosine functions for each angle measure.

B, given cos 2B = 1/8 , 540° < 2B < 720°

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Textbook Question

Let csc x = -3. Find all possible values of (sin x + cos x)/sec x.

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