Ch. 5 - Trigonometric Identities

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 5 - Trigonometric Identities

Problem 5.RE.36

Chapter 6, Problem 5.RE.36

Find values of the sine and cosine functions for each angle measure.
2y, given sec y = -5/3, sin y > 0

Verified step by step guidance

Identify the given information: \( \sec y = -\frac{5}{3} \) and \( \sin y > 0 \). Recall that \( \sec y = \frac{1}{\cos y} \), so first find \( \cos y \) by taking the reciprocal of \( \sec y \).

Determine the quadrant of angle \( y \) using the signs of \( \cos y \) and \( \sin y \). Since \( \sec y = -\frac{5}{3} \), \( \cos y \) is negative, and \( \sin y > 0 \) means sine is positive. Use this to identify the correct quadrant.

Use the Pythagorean identity \( \sin^2 y + \cos^2 y = 1 \) to find \( \sin y \). Substitute the value of \( \cos y \) found in step 1 into the identity and solve for \( \sin y \), choosing the positive root because \( \sin y > 0 \).

Find the values of \( \sin 2y \) and \( \cos 2y \) using the double-angle formulas: \( \sin 2y = 2 \sin y \cos y \) and \( \cos 2y = \cos^2 y - \sin^2 y \). Substitute the values of \( \sin y \) and \( \cos y \) obtained earlier.

Express the final answers for \( \sin 2y \) and \( \cos 2y \) in simplified form, leaving the expressions in terms of fractions or radicals without calculating decimal approximations.

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Key Concepts

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

Reciprocal Trigonometric Functions

The secant function (sec) is the reciprocal of the cosine function, meaning sec y = 1/cos y. Knowing sec y allows us to find cos y by taking the reciprocal, which is essential for determining sine and cosine values for related angles.

Sign of Trigonometric Functions in Quadrants

The sign of sine and cosine depends on the quadrant where the angle lies. Given sin y > 0 and sec y = -5/3, we deduce the quadrant of y, which helps determine the correct signs of sine and cosine values for y and related angles like 2y.

Double-Angle Formulas

Double-angle formulas express trigonometric functions of 2y in terms of functions of y. For sine and cosine, these formulas are sin 2y = 2 sin y cos y and cos 2y = cos^2 y - sin^2 y, enabling calculation of sine and cosine for 2y once values for y are known.

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Double Angle Identities

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Textbook Question

Use the given information to cos(x - y).

cos x = 2/9, sin y = -1/2, x in quadrant IV, y in quadrant III

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Textbook Question

For each expression in Column I, use an identity to choose an expression from Column II with the same value. Choices may be used once, more than once, or not at all.

cos 75°

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Textbook Question

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

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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

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