Textbook Question
Find values of the sine and cosine functions for each angle measure.
2θ, given cos θ = -12/13 and sin θ > 0
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 8
Match each expression in Column I with its equivalent expression in Column II.
(tan (π/3) - tan (π/4))/(1 + tan (π/3) tan (π/4))
Verified step by step guidance1
Recognize that the given expression \( \frac{\tan(\frac{\pi}{3}) - \tan(\frac{\pi}{4})}{1 + \tan(\frac{\pi}{3}) \tan(\frac{\pi}{4})} \) matches the formula for the tangent of a difference of two angles: \( \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \).
Identify the angles \( A = \frac{\pi}{3} \) and \( B = \frac{\pi}{4} \) in the expression, so the expression simplifies to \( \tan\left(\frac{\pi}{3} - \frac{\pi}{4}\right) \).
Calculate the difference of the angles inside the tangent function: \( \frac{\pi}{3} - \frac{\pi}{4} = \frac{4\pi}{12} - \frac{3\pi}{12} = \frac{\pi}{12} \).
Rewrite the original expression as \( \tan\left(\frac{\pi}{12}\right) \), which is the equivalent expression in Column II.
If needed, recall or use the exact value or approximation of \( \tan\left(\frac{\pi}{12}\right) \) to verify the equivalence with the expressions in Column II.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Tangent Function and Its Values
The tangent function relates an angle in a right triangle to the ratio of the opposite side over the adjacent side. Key values such as tan(π/3) = √3 and tan(π/4) = 1 are often used in trigonometric calculations and simplifications.
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Sine, Cosine, & Tangent of 30°, 45°, & 60°
Tangent Difference Formula
The tangent difference formula states that (tan A - tan B) / (1 + tan A tan B) equals tan(A - B). This identity is essential for simplifying expressions involving differences of tangents into a single tangent function.
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4:47Sum and Difference of Tangent
Angle Subtraction in Radians
Understanding how to subtract angles measured in radians, such as π/3 - π/4, is crucial for applying trigonometric identities correctly. This involves finding a common denominator and simplifying the resulting angle before evaluating the tangent.
Recommended video:
5:04Converting between Degrees & Radians
Related Practice
Textbook Question
Match each expression in Column I with its value in Column II.
(2 tan (π/3))/(1 - tan² (π/3))
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Textbook Question
Find the exact value of each expression.
sin 255°
Textbook Question
Find values of the sine and cosine functions for each angle measure.
2x, given tan x = 5/3 and sin x < 0
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Textbook Question
For each expression in Column I, choose the expression from Column II that completes an identity.
6. sec² 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
Textbook Question
Match each expression in Column I with its equivalent expression in Column II.
sin 60° cos 45° - cos 60° sin 45°
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