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
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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.30c
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
Verified step by step guidance1
Identify the given information and the goal: We need to find \(\tan(x + y)\) given \(\sin y = -\frac{2}{3}\), \(\cos x = -\frac{1}{5}\), with \(x\) in quadrant II and \(y\) in quadrant III.
Recall the formula for the tangent of a sum: \(\tan(x + y) = \frac{\tan x + \tan y}{1 - \tan x \tan y}\).
Find \(\sin x\) using the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\). Since \(\cos x = -\frac{1}{5}\) and \(x\) is in quadrant II (where sine is positive), calculate \(\sin x = +\sqrt{1 - \left(-\frac{1}{5}\right)^2}\).
Find \(\cos y\) using the Pythagorean identity \(\sin^2 y + \cos^2 y = 1\). Since \(\sin y = -\frac{2}{3}\) and \(y\) is in quadrant III (where cosine is negative), calculate \(\cos y = -\sqrt{1 - \left(-\frac{2}{3}\right)^2}\).
Calculate \(\tan x = \frac{\sin x}{\cos x}\) and \(\tan y = \frac{\sin y}{\cos y}\), then substitute these values into the formula \(\tan(x + y) = \frac{\tan x + \tan y}{1 - \tan x \tan y}\) to find the expression for \(\tan(x + y)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Identities for Sum of Angles
The tangent of a sum, tan(x + y), can be found using the identity tan(x + y) = (tan x + tan y) / (1 - tan x * tan y). This formula allows combining the tangents of individual angles to find the tangent of their sum.
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2:25
Verifying Identities with Sum and Difference Formulas
Determining Signs of Trigonometric Functions in Quadrants
The signs of sine, cosine, and tangent depend 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. Correct sign assignment is crucial for accurate calculations.
Recommended video:
6:04Introduction to Trigonometric Functions
Finding Missing Trigonometric Ratios Using Pythagorean Identity
Given one trigonometric ratio (like sin y or cos x), the other ratios can be found using the Pythagorean identity sin²θ + cos²θ = 1. This helps determine unknown values such as cos y or sin x, essential for calculating tan x and tan y.
Recommended video:
6:25Pythagorean Identities
Related Practice
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
846
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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.
-sin 35°
617
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Textbook Question
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
747
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Textbook Question
Use the given information to find each of the following.
sin 2x, given sin x = 0.6, π/2 < y < π
828
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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°)
741
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