Textbook Question
Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.
(1 - cos 2x)/sin 2x
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.30d
Use the given information to find the quadrant of 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.
Recall the signs of sine and cosine in each quadrant: In quadrant II, sine is positive and cosine is negative; in quadrant III, sine and cosine are both negative.
Find \(\cos y\) using the Pythagorean identity \(\sin^2 y + \cos^2 y = 1\). Substitute \(\sin y = -\frac{2}{3}\) and solve for \(\cos y\), considering the sign of cosine in quadrant III (which is negative).
Find \(\sin x\) using the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\). Substitute \(\cos x = -\frac{1}{5}\) and solve for \(\sin x\), considering the sign of sine in quadrant II (which is positive).
Use the angle sum formulas for sine and cosine: \(\sin(x+y) = \sin x \cos y + \cos x \sin y\) and \(\cos(x+y) = \cos x \cos y - \sin x \sin y\). Determine the signs of \(\sin(x+y)\) and \(\cos(x+y)\) to identify the quadrant of \(x + y\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Ratios and Signs in Quadrants
Trigonometric functions like sine and cosine have specific signs depending on the quadrant of the angle. In quadrant II, sine is positive and cosine is negative; in quadrant III, both sine and cosine are negative. Understanding these sign conventions helps determine the values and behavior of angles.
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Quadratic Formula
Determining Angle Quadrants from Trigonometric Values
Given the value and sign of a trigonometric function, one can deduce the possible quadrant(s) of the angle. For example, sin y = -2/3 indicates y is in quadrant III or IV, but since y is given in quadrant III, this confirms the angle's location. This aids in correctly identifying angle sums.
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5:30Trig Values in Quadrants II, III, & IV
Sum of Angles and Quadrant Determination
The sum of two angles x and y can be analyzed by considering their individual quadrants and trigonometric values. By using angle addition formulas or sign rules, one can determine the quadrant of x + y, which depends on the combined angle's sine and cosine signs.
Recommended video:
6:36Quadratic Formula
Related Practice
Textbook Question
Use the given information to find tan(x + y).
sin y = - 2/3, cos x = -1/5, x in quadrant II, y in quadrant III
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Textbook Question
Use the given information to find sin(x + y), cos(x - y), tan(x + y), and the quadrant of x + y.
sin x = 3/5, cos y = 24/25, x in quadrant I, y in quadrant IV
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
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)
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