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.
csc θ - sin θ
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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.32b
Use the given information to cos(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 given information: \(\cos x = \frac{2}{9}\), \(\sin y = -\frac{1}{2}\), with \(x\) in quadrant IV and \(y\) in quadrant III.
Find \(\sin x\) using the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\). Since \(x\) is in quadrant IV, where sine is negative, calculate \(\sin x = -\sqrt{1 - \left(\frac{2}{9}\right)^2}\).
Find \(\cos y\) using the Pythagorean identity \(\sin^2 y + \cos^2 y = 1\). Since \(y\) is in quadrant III, where cosine is negative, calculate \(\cos y = -\sqrt{1 - \left(-\frac{1}{2}\right)^2}\).
Use the cosine difference formula: \(\cos(x - y) = \cos x \cos y + \sin x \sin y\).
Substitute the values of \(\cos x\), \(\cos y\), \(\sin x\), and \(\sin y\) into the formula and simplify the expression to find \(\cos(x - y)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Identities - Cosine of a Difference
The cosine of a difference formula states that cos(x - y) = cos x cos y + sin x sin y. This identity allows us to express the cosine of the difference of two angles in terms of the sines and cosines of the individual angles, which is essential for solving the problem.
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Sum and Difference of Sine & Cosine
Determining Signs of Trigonometric Functions by Quadrant
The signs of sine and cosine depend on the quadrant of the angle. In quadrant IV, cosine is positive and sine is negative; in quadrant III, both sine and cosine are negative. Knowing the quadrant helps assign correct signs to sin x and cos y when they are not directly given.
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6:04Introduction to Trigonometric Functions
Using Pythagorean Identity to Find Missing Values
The Pythagorean identity, sin²θ + cos²θ = 1, allows calculation of an unknown sine or cosine value when the other is known. For example, if cos x is given, sin x can be found by sin x = ±√(1 - cos²x), with the sign determined by the quadrant.
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6:25Pythagorean Identities
Related Practice
Textbook Question
Find values of the sine and cosine functions for each angle measure.
2y, given sec y = -5/3, sin y > 0
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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°
Textbook Question
Use the given information to find each of the following.
sin A/2, given cos A/2 = - 3, 90° < A < 180°
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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.
csc² θ + sec² θ
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Textbook Question
Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.
(cos x sin 2x)/1 + cos 2x)
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