Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 7: Q. 42 (page 504)

Establish each identity
$\frac{\sin (3 胃) \cos 胃 - \sin 胃 \cos (3 胃)}{\sin (2 胃)} = 1$

Short Answer

Expert verified

The derivation of the equation $\frac{\sin (3 胃) \cos 胃 - \sin 胃 \cos (3 胃)}{\sin (2 胃)} = 1$ is

$\frac{\sin (3 胃) \cos 胃 - \sin 胃 \cos (3 胃)}{\sin (2 胃)} = \frac{(3 \sin 胃 - 4 \sin^{3} 胃) \cos 胃 - \sin 胃 (4 \cos^{3} 胃 - 3 \cos 胃)}{\sin (2 胃)} = \frac{6 \sin 胃 \cos 胃 - 4 \sin 胃 \cos 胃 (\sin^{2} 胃 + \cos^{2} 胃)}{\sin (2 胃)} = \frac{2 \sin 胃 \cos 胃}{\sin (2 胃)} = \frac{2 \sin 胃 \cos 胃}{2 \sin 胃 \cos 胃} = 1$

Step by step solution

Step 1. Given data

The given equation for derivation is

$\frac{\sin (3 胃) \cos 胃 - \sin 胃 \cos (3 胃)}{\sin (2 胃)} = 1$

Step 2. Use of triple angle formula

Take left-hand side expression and use triple angle formula

$\frac{\sin (3 胃) \cos 胃 - \sin 胃 \cos (3 胃)}{\sin (2 胃)} = \frac{(3 \sin 胃 - 4 \sin^{3} 胃) \cos 胃 - \sin 胃 (4 \cos^{3} 胃 - 3 \cos 胃)}{\sin (2 胃)} = \frac{3 \sin 胃 \cos 胃 - 4 \sin^{3} 胃 \cos 胃 - 4 \sin 胃 \cos^{3} 胃 + 3 \cos 胃 \sin 胃}{\sin (2 胃)} = \frac{6 \sin 胃 \cos 胃 - 4 \sin 胃 \cos 胃 (\sin^{2} 胃 + \cos^{2} 胃)}{\sin (2 胃)} = \frac{6 \sin 胃 \cos 胃 - 4 \sin 胃 \cos 胃 (1)}{\sin (2 胃)} = \frac{2 \sin 胃 \cos 胃}{\sin (2 胃)}$

Step 3. Use of double angle formula

Apply double angle formula

$\frac{\sin (3 胃) \cos 胃 - \sin 胃 \cos (3 胃)}{\sin (2 胃)} = \frac{2 \sin 胃 \cos 胃}{\sin (2 胃)} \frac{\sin (3 胃) \cos 胃 - \sin 胃 \cos (3 胃)}{\sin (2 胃)} = \frac{2 \sin 胃 \cos 胃}{2 \sin 胃 \cos 胃} \frac{\sin (3 胃) \cos 胃 - \sin 胃 \cos (3 胃)}{\sin (2 胃)} = 1$

The left-hand side expression is equal to the right-hand side expression

So identity is stabilised

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91影视

Establish each identity
$\frac{\sin (3 胃) \cos 胃 - \sin 胃 \cos (3 胃)}{\sin (2 胃)} = 1$

Short Answer

Step by step solution

Step 1. Given data

Step 2. Use of triple angle formula

Step 3. Use of double angle formula

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91影视

Establish each identitysin3胃cos胃-sin胃cos3胃sin2胃=1

Short Answer

Step by step solution

Step 1. Given data

Step 2. Use of triple angle formula

Step 3. Use of double angle formula

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Discrete Mathematics

Theoretical and Mathematical Physics

Calculus

Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Establish each identity
$\frac{\sin (3 胃) \cos 胃 - \sin 胃 \cos (3 胃)}{\sin (2 胃)} = 1$