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Ch. 5 - Trigonometric Identities
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 5 - Trigonometric IdentitiesProblem 46
Chapter 6, Problem 46

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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Recall the cosine difference identity: \(\cos(A - B) = \cos A \cos B + \sin A \sin B\).
Apply the identity to the expression \(\cos(\theta - 270^\circ)\) by letting \(A = \theta\) and \(B = 270^\circ\).
Write the expression as \(\cos \theta \cos 270^\circ + \sin \theta \sin 270^\circ\).
Substitute the known values: \(\cos 270^\circ = 0\) and \(\sin 270^\circ = -1\).
Simplify the expression to get a trigonometric function involving only \(\theta\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Cosine of a Difference Identity

The cosine of a difference identity states that cos(A - B) = cos A cos B + sin A sin B. This formula allows you to express the cosine of the difference between two angles in terms of the cosines and sines of the individual angles, which is essential for rewriting expressions like cos(θ - 270°).
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Sum and Difference of Sine & Cosine

Values of Trigonometric Functions at Special Angles

Certain angles, such as 0°, 90°, 180°, 270°, and 360°, have known sine and cosine values. For example, cos 270° = 0 and sin 270° = -1. Knowing these values helps simplify expressions involving these angles, enabling you to rewrite cos(θ - 270°) in terms of cos θ and sin θ.
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Introduction to Trigonometric Functions

Simplification of Trigonometric Expressions

After applying identities and substituting known values, simplifying the resulting expression is crucial. This involves combining like terms and reducing the expression to a single trigonometric function of θ, making it easier to interpret or use in further calculations.
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Simplifying Trig Expressions
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

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

Simplify each expression. See Example 4.

tan 34°/2(1 - tan² 34°)

Textbook Question

Verify that each equation is an identity.

cot² (x/2) = (1 + cos x)²/(sin² x)

Textbook Question

Write each function as an expression involving functions of θ or x alone. See Example 2.

sin(π + x)