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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 60. (page 458)

In Problem, write each trigonometric expression as an algebraic expression in $u$ .
$\tan (\cos^{- 1} u)$

Short Answer

Expert verified

Trigonometric expression $\tan (c o s^{- 1} u)$ as an algebraic expression in $u$ is given by $\begin{array}{rcl} \frac{\sqrt{1 - u^{2}}}{u} \end{array}$ .

Step by step solution

01

Step 1. Given information

An expression is $\tan (\cos^{- 1} u)$ .

02

Step 2. Value of the expression in terms of u

Consider the expression $\tan (\cos^{- 1} u)$ .

Let $\cos^{- 1} u = x$

So, $u = \cos x$

Consider the following figure:

Side opposite to angle $x = \sqrt{1 - u^{2}}$

localid="1646562158872" $\begin{array}{rcl} \tan x & = & \frac{\sqrt{1 - u^{2}}}{u} \\ x & = & \tan^{- 1} (\frac{\sqrt{1 - u^{2}}}{u}) \end{array}$

Therefore,

$\begin{array}{rcl} \tan (\cos^{- 1} u) & = & \tan [\tan^{- 1} (\frac{\sqrt{1 - u^{2}}}{u})] \\ = & \frac{\sqrt{1 - u^{2}}}{u} \end{array}$

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Most popular questions from this chapter

In Problems 9鈥�36, find the exact value of each expression.

$\csc [\cos^{- 1} (- \frac{\sqrt{3}}{2})]$

$\tan^{2} (胃) - \sec^{2} (胃)$

Establish each identity.

$\frac{1 + \sin 胃}{1 - \sin 胃} = {(s e c 胃 + \tan 胃)}^{2}$

In Problems 9鈥�36, find the exact value of each expression.

$\tan [\cos^{- 1} (- \frac{\sqrt{3}}{2})]$

Establish each identity.

$\frac{\sin 胃 + \cos 胃}{\cos 胃} - \frac{\sin 胃 - \cos 胃}{\sin 胃} = s e c 胃 c s c 胃$

See all solutions

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