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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 7: Q. 102 (page 498)

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

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Given information.

Consider the given question,

$x = 2 \tan 胃$

From the double-angle formula,

$\cos (2 胃) = 2 \cos^{2} 胃 - 1$

From inverse trigonometry,

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

02

Step 2. Use inverse trigonometry and solve the equation.

From inverse trigonometry,

$胃 = \tan^{- 1} \frac{x}{2}$

Then,

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

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Most popular questions from this chapter

A rectangle is inscribed in a semicircle of radius $1$ . See the illustration.

Part (a): Express the area A of the rectangle as a function of the angle $胃$ shown in the illustration.

Part (b): Show that $A (胃) = \sin (2 胃)$ .

Part (c): Find the angle $胃$ that results in the largest area A.

Part (d): Find the dimensions of this largest rectangle.

(a) $\sin \frac{蟺}{4} 路 \cos \frac{蟺}{3} =______$

(b) $\tan \frac{蟺}{4} - \sin \frac{蟺}{6} =________$

If $\tan 胃 = a \tan \frac{胃}{3}$ , express $\tan \frac{胃}{3}$ in terms of a.

Express the product $\cos (4 胃) \cos (2 胃)$ as a sum containing only sines or only cosines.

Establish each identity.

${(a \sin 胃 + b \cos 胃)}^{2} + {(a \cos 胃 - b \sin 胃)}^{2} = a^{2} + b^{2}$

See all solutions

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