Textbook Question
Simplify each expression. See Example 4.
cos² π/8 - 1/2
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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 42
Find one value of θ or x that satisfies each of the following.
cot(θ - 10°) = tan(2θ - 20°)
Verified step by step guidance1
Recall the identity that relates cotangent and tangent: \(\cot A = \tan(90^\circ - A)\). Use this to rewrite \(\cot(\theta - 10^\circ)\) as \(\tan\big(90^\circ - (\theta - 10^\circ)\big)\).
Rewrite the equation \(\cot(\theta - 10^\circ) = \tan(2\theta - 20^\circ)\) as \(\tan\big(90^\circ - (\theta - 10^\circ)\big) = \tan(2\theta - 20^\circ)\).
Simplify the expression inside the tangent on the left side: \(90^\circ - (\theta - 10^\circ) = 90^\circ - \theta + 10^\circ = 100^\circ - \theta\).
Set the arguments of the tangent functions equal to each other, considering the periodicity of tangent: \(100^\circ - \theta = 2\theta - 20^\circ + k \times 180^\circ\), where \(k\) is any integer.
Solve the resulting linear equation for \(\theta\) to find one or more values that satisfy the original equation.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Relationship Between Cotangent and Tangent
Cotangent and tangent are reciprocal trigonometric functions, where cot(α) = 1/tan(α). Recognizing that cot(θ - 10°) can be rewritten as tan(90° - (θ - 10°)) helps transform the equation into a more solvable form by using complementary angle identities.
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Introduction to Cotangent Graph
Trigonometric Equation Solving Techniques
Solving equations involving trigonometric functions often requires using identities, algebraic manipulation, and understanding periodicity. Equating angles or their related expressions and considering the periodic nature of tangent and cotangent functions is essential to find valid solutions.
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4:34How to Solve Linear Trigonometric Equations
Angle Identities and Complementary Angles
The identity tan(90° - x) = cot(x) links tangent and cotangent through complementary angles. Applying this identity allows rewriting cot(θ - 10°) as tan(100° - θ), enabling the equation to be expressed in terms of tangent functions for easier comparison and solution.
Recommended video:
3:35Intro to Complementary & Supplementary Angles
Related Practice
Textbook Question
Write each function as an expression involving functions of θ or x alone. See Example 2.
tan(180° + θ)
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Textbook Question
Write each function as an expression involving functions of θ or x alone. See Example 2.
tan (π/4 + x)
731
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Textbook Question
Simplify each expression.
±√[(1 - cos 5A)/(1 + cos 5A)]
744
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Textbook Question
Simplify each expression. See Example 4.
1 - 2 sin² 22 ½°
601
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Textbook Question
Use the identities for the cosine of a sum or difference to write each expression as a trigonometric function of θ alone.
cos(90° - θ)
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