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.42
Use the given information to find each of the following.
sin 2x, given sin x = 0.6, π/2 < y < π
Verified step by step guidance1
Identify the given information: \( \sin x = 0.6 \) and the angle \( x \) lies in the interval \( \frac{\pi}{2} < x < \pi \), which means \( x \) is in the second quadrant.
Recall the double-angle formula for sine: \( \sin 2x = 2 \sin x \cos x \). To find \( \sin 2x \), we need both \( \sin x \) and \( \cos x \).
Use the Pythagorean identity to find \( \cos x \): \( \cos x = \pm \sqrt{1 - \sin^2 x} \). Since \( x \) is in the second quadrant, where cosine is negative, choose the negative root.
Calculate \( \cos x = -\sqrt{1 - (0.6)^2} \) without simplifying the square root fully, just set up the expression.
Substitute \( \sin x = 0.6 \) and the expression for \( \cos x \) into the double-angle formula \( \sin 2x = 2 \times 0.6 \times \cos x \) to express \( \sin 2x \) in terms of known values.
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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 Sine
The double-angle identity for sine states that sin(2x) = 2 sin(x) cos(x). This formula allows you to find the sine of twice an angle if you know the sine and cosine of the original angle.
Recommended video:
05:06
Double Angle Identities
Using the Pythagorean Identity to Find Cosine
Given sin(x), you can find cos(x) using the Pythagorean identity: sin²(x) + cos²(x) = 1. Rearranging gives cos(x) = ±√(1 - sin²(x)). The sign depends on the quadrant where x lies.
Recommended video:
6:25Pythagorean Identities
Determining the Sign of Trigonometric Functions Based on Quadrants
The sign of sine and cosine depends on the quadrant of the angle. Since π/2 < x < π (second quadrant), sin(x) is positive and cos(x) is negative. This helps determine the correct sign for cos(x) when using the Pythagorean identity.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
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 300°
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.
-sin 35°
Textbook Question
Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.
cot(-θ)/sec(-θ)
1
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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(-55°)