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(θ - 270°)
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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 44
Simplify each expression. See Example 4.
tan 34°/2(1 - tan² 34°)
Verified step by step guidance1
Recognize that the expression resembles the double-angle formula for tangent. Recall the formula: \(\tan(2\theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta}\).
Compare the given expression \(\frac{\tan 34^\circ}{2(1 - \tan^2 34^\circ)}\) with the double-angle formula. Notice that the numerator and denominator are arranged differently.
Rewrite the given expression to see if it can be manipulated into the form of the double-angle formula. For example, consider multiplying numerator and denominator appropriately or factoring to match the formula's structure.
Use the double-angle identity to express \(\tan 68^\circ\) in terms of \(\tan 34^\circ\) and relate it back to the given expression.
Conclude the simplification by substituting the equivalent double-angle expression, which will simplify the original expression into a single tangent function or a simpler trigonometric expression.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Tangent Double-Angle Identity
The tangent double-angle identity expresses tan(2θ) in terms of tan(θ): tan(2θ) = 2 tan(θ) / (1 - tan²(θ)). This identity helps simplify expressions involving tangent of multiple angles by rewriting them in terms of a single angle.
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05:06
Double Angle Identities
Algebraic Simplification of Trigonometric Expressions
Simplifying trigonometric expressions often involves factoring, combining like terms, and applying identities. Recognizing patterns such as the double-angle formula allows rewriting complex fractions into simpler forms.
Recommended video:
6:36Simplifying Trig Expressions
Evaluating Trigonometric Functions at Specific Angles
Understanding how to evaluate or manipulate trigonometric functions at given angles, like 34°, is essential. This includes knowing approximate values or using identities to rewrite expressions for easier calculation or simplification.
Recommended video:
3:48Evaluate Composite Functions - Special Cases
Related Practice
Textbook Question
Write each function as an expression involving functions of θ or x alone. See Example 2.
tan(180° + θ)
Textbook Question
Find one value of θ or x that satisfies each of the following.
cot(θ - 10°) = tan(2θ - 20°)
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Textbook Question
Simplify each expression. See Example 4.
⅛ sin 29.5° cos 29.5°
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Textbook Question
Simplify each expression.
±√[(1 - cos (3θ/5))/2]
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)