Textbook Question
Match each expression in Column I with its value in Column II.
(2 tan 15°)/(1 - tan² 15°)
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 4
Match each expression in Column I with its value in Column II.
cos² (π/6) - sin² (π/6)
Verified step by step guidance1
Recall the trigonometric identity for the difference of squares of cosine and sine: \(\cos^2 x - \sin^2 x = \cos 2x\).
Apply this identity to the given expression by substituting \(x = \frac{\pi}{6}\), so the expression becomes \(\cos 2 \times \frac{\pi}{6} = \cos \frac{\pi}{3}\).
Evaluate \(\cos \frac{\pi}{3}\) by recalling the unit circle values or special angles, where \(\cos \frac{\pi}{3} = \frac{1}{2}\).
Match the original expression \(\cos^2 \left(\frac{\pi}{6}\right) - \sin^2 \left(\frac{\pi}{6}\right)\) with the value \(\frac{1}{2}\) from Column II.
Verify the result by optionally calculating \(\cos^2 \left(\frac{\pi}{6}\right)\) and \(\sin^2 \left(\frac{\pi}{6}\right)\) separately and subtracting to confirm the value.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Pythagorean Identity
The Pythagorean identity states that for any angle θ, sin²θ + cos²θ = 1. This fundamental relationship helps simplify expressions involving sine and cosine squares by relating them to each other.
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Pythagorean Identities
Double-Angle Formula for Cosine
The double-angle formula for cosine states that cos(2θ) = cos²θ - sin²θ. This formula allows rewriting expressions like cos²θ - sin²θ in terms of a single cosine function with double the angle, simplifying evaluation.
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05:06Double Angle Identities
Exact Values of Trigonometric Functions at Special Angles
Certain angles, such as π/6 (30°), have known exact sine and cosine values: sin(π/6) = 1/2 and cos(π/6) = √3/2. Using these values enables precise calculation of trigonometric expressions without a calculator.
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Related Practice
Textbook Question
Verify that each equation is an identity.
[(sec θ - tan θ)² + 1]/(sec θ csc θ - tan θ csc θ) = 2 tan θ
Textbook Question
Find the exact value of each expression. (Do not use a calculator.)
cos(-15°)
Textbook Question
Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
cot² θ(1 + tan² θ)