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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 6
Chapter 6, Problem 6

Match each expression in Column I with its equivalent expression in Column II.
Matching exercise with trigonometric expressions in two columns involving sine, cosine, and tangent functions with angles in degrees and radians.
sin 60° cos 45° - cos 60° sin 45°

Verified step by step guidance
1
Recognize that the expression given is of the form \(\sin A \cos B - \cos A \sin B\), which matches the sine difference identity.
Recall the sine difference identity: \(\sin A \cos B - \cos A \sin B = \sin (A - B)\).
Identify the angles: \(A = 60^\circ\) and \(B = 45^\circ\).
Apply the identity to rewrite the expression as \(\sin (60^\circ - 45^\circ)\).
Simplify the angle inside the sine function to get \(\sin 15^\circ\), which is the equivalent expression.

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Key Concepts

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

Sine of a Difference Formula

The sine of a difference of two angles, sin(A - B), is given by sin A cos B minus cos A sin B. This identity allows the expression sin 60° cos 45° - cos 60° sin 45° to be recognized as sin(60° - 45°), simplifying the evaluation.
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Verifying Identities with Sum and Difference Formulas

Basic Trigonometric Values

Knowing the exact values of sine and cosine for common angles like 45° and 60° is essential. For example, sin 45° = cos 45° = √2/2, sin 60° = √3/2, and cos 60° = 1/2. These values help in simplifying and verifying trigonometric expressions.
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Fundamental Trigonometric Identities

Trigonometric Expression Matching

Matching expressions involves recognizing equivalent forms using identities and simplifications. Understanding how to rewrite expressions using formulas like angle sum/difference helps in pairing expressions from different columns correctly.
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Simplifying Trig Expressions
Related Practice
Textbook Question

Find values of the sine and cosine functions for each angle measure.

2θ, given cos θ = -12/13 and sin θ > 0

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

Use the given information to find cos(x - y).

sin y = - 2/3, cos x = -1/5, x in quadrant II, y in quadrant III

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.

(tan (π/3) - tan (π/4))/(1 + tan (π/3) tan (π/4))

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Textbook Question

Use the given information to find sin(x + y).

sin y = - 2/3 , cos x = - 1/5, x in quadrant II, y in quadrant III