Book edition 6th

Author(s) Sullivan

Pages 1200 pages

ISBN 9780321795465

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Short Answer

If $x = 2 \tan θ$ , express $\cos (2 θ)$ as a function of x.

On expressing $\cos (2 θ)$ as a function of x is $\cos (2 θ) = \frac{4 - x^{2}}{4 + x^{2}}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

Consider the given question,

$x = 2 \tan θ$

From the double-angle formula,

$\cos (2 θ) = 2 \cos^{2} θ - 1$

From inverse trigonometry,

$θ = \tan^{- 1} \frac{x}{2} = \sin^{- 1} (\frac{x}{\sqrt{4 + x^{2}}}) = c o s^{- 1} (\frac{2}{\sqrt{4 + x^{2}}})$

Step 2. Use inverse trigonometry and solve the equation.

From inverse trigonometry,

$θ = \tan^{- 1} \frac{x}{2}$

Then,

$θ = \tan^{- 1} \frac{x}{2} \cos (2 θ) = 2 \cos^{2} (\tan^{- 1} \frac{x}{2}) - 1 \cos (2 θ) = 2 \cos^{2} (\cos^{- 1} (\frac{2}{\sqrt{4 + x^{2}}})) - 1 \cos (2 θ) = 2 {(\frac{2}{\sqrt{4 + x^{2}}})}^{2} - 1 \cos (2 θ) = \frac{4 - x^{2}}{4 + x^{2}}$

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Precalculus Enhanced with Graphing Utilities

Answers without the blur.

Short Answer

If $x = 2 \tan θ$ , express $\cos (2 θ)$ as a function of x.

Step by Step Solution

Step 1. Given information.

Step 2. Use inverse trigonometry and solve the equation.

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Precalculus Enhanced with Graphing Utilities

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Short Answer

If x=2tanθ, express cos2θ as a function of x.

Step by Step Solution

Step 1. Given information.

Step 2. Use inverse trigonometry and solve the equation.

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If $x = 2 \tan θ$ , express $\cos (2 θ)$ as a function of x.