Textbook Question
For each expression in Column I, choose the expression from Column II that completes an identity.
2. csc 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
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.40
Use the given information to find each of the following.
sin y, given cos 2y = -1/3 , π/2 < y < π
Verified step by step guidance1
Recall the double-angle identity for cosine: \(\cos(2y) = 2\cos^2(y) - 1\).
Use the given value \(\cos(2y) = -\frac{1}{3}\) and substitute it into the identity: \(-\frac{1}{3} = 2\cos^2(y) - 1\).
Solve the equation for \(\cos^2(y)\): add 1 to both sides to get \(\frac{2}{3} = 2\cos^2(y)\), then divide both sides by 2 to find \(\cos^2(y) = \frac{1}{3}\).
Find \(\cos(y)\) by taking the square root: \(\cos(y) = \pm \sqrt{\frac{1}{3}} = \pm \frac{1}{\sqrt{3}}\). Determine the correct sign of \(\cos(y)\) using the interval \(\frac{\pi}{2} < y < \pi\), where cosine is negative.
Use the Pythagorean identity \(\sin^2(y) + \cos^2(y) = 1\) to find \(\sin(y)\): substitute \(\cos^2(y) = \frac{1}{3}\), then solve for \(\sin(y)\), considering the sign of \(\sin(y)\) in the given interval.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Double-Angle Identity for Cosine
The double-angle identity states that cos(2y) = 2cos²(y) - 1 or cos(2y) = 1 - 2sin²(y). This identity allows us to express cos(2y) in terms of sin(y) or cos(y), which is essential for finding sin(y) when cos(2y) is known.
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Double Angle Identities
Sign of Trigonometric Functions in Quadrants
The value of sin(y) depends on the quadrant where angle y lies. Since π/2 < y < π, y is in the second quadrant where sine is positive and cosine is negative. This information helps determine the correct sign of sin(y) after calculation.
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Pythagorean Identity
The Pythagorean identity, sin²(y) + cos²(y) = 1, relates sine and cosine of the same angle. It is useful for finding sin(y) once cos(y) is determined or vice versa, ensuring the values satisfy this fundamental trigonometric relationship.
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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
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
Work each problem.
Find the exact values of sin x, cos x, and tan x, for x = π/12 , using
a. difference identities
b. half-angle identities.
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(-55°)
Textbook Question
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