Textbook Question
Use the given information to find each of the following.
sin y, given cos 2y = -1/3 , π/2 < y < π
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Ch. 5 - Trigonometric Identities
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
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 guidance1
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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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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Related Practice
Textbook Question
Use the given information to find the quadrant of x + y.
cos x = 2/9, sin y = -1/2, x in quadrant IV, y in quadrant III
Textbook Question
Use the given information to find cos(x - y).
sin y = - 2/3, cos x = -1/5, x in quadrant II, y in quadrant III
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.
sin 35°
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
Textbook Question
Match each expression in Column I with its equivalent expression in Column II.
sin 60° cos 45° - cos 60° sin 45°
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