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.32d
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
Verified step by step guidance1
Identify the signs of sine and cosine for angles x and y based on their quadrants: Since x is in quadrant IV, cos x is positive and sin x is negative; since y is in quadrant III, sin y is negative and cos y is also negative.
Use the Pythagorean identity to find sin x: Since \( \cos x = \frac{2}{9} \) and x is in quadrant IV where sine is negative, calculate \( \sin x = -\sqrt{1 - \left(\frac{2}{9}\right)^2} \).
Use the Pythagorean identity to find cos y: Since \( \sin y = -\frac{1}{2} \) and y is in quadrant III where cosine is negative, calculate \( \cos y = -\sqrt{1 - \left(-\frac{1}{2}\right)^2} \).
Use the cosine addition formula to find \( \cos(x + y) \): \[ \cos(x + y) = \cos x \cos y - \sin x \sin y \]. Substitute the values found for \( \cos x, \cos y, \sin x, \sin y \).
Determine the quadrant of \( x + y \) by analyzing the sign of \( \cos(x + y) \) and \( \sin(x + y) \) (which can be found using the sine addition formula \( \sin(x + y) = \sin x \cos y + \cos x \sin y \)). The signs of sine and cosine will indicate the quadrant.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Ratios and Quadrants
Trigonometric ratios like sine and cosine vary in sign depending on the quadrant of the angle. Knowing the quadrant helps determine whether these values are positive or negative, which is essential for solving problems involving angle sums or differences.
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Quadratic Formula
Sum of Angles and Quadrant Determination
The sum of two angles can be located in a specific quadrant based on the individual angles' quadrants and their trigonometric values. Understanding how to combine angles and analyze their sum's position on the unit circle is key to identifying the correct quadrant.
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Using Pythagorean Identity to Find Missing Ratios
When one trigonometric ratio is given, the Pythagorean identity (sin²θ + cos²θ = 1) allows calculation of the other ratio. This is crucial when determining the sine or cosine of angles to find the quadrant of their sum.
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Related Practice
Textbook Question
Use the given information to find each of the following.
sin y, given cos 2y = -1/3 , π/2 < y < π
3
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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, 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