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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. 85 (page 487)

Write each trigonometric expression as an algebraic expression containing $u$ and $v$ . Give the restrictions required on $u$ and $v$ .
role="math" localid="1646564067411" $\cos (\cos^{- 1} u + \sin^{- 1} v)$

Short Answer

Expert verified

We can write $\cos (\cos^{- 1} u + \sin^{- 1} v)$ as $u \sqrt{1 - v^{2}} - v \sqrt{1 - u^{2}}$ , where $- 1 鈮� u 鈮� 1, - 1 鈮� v 鈮� 1$

Step by step solution

01

Step 1. Given information

The given expression is $\cos (\cos^{- 1} u + \sin^{- 1} v)$

02

Step 2. Solve the expression

Let $伪 = \cos^{- 1} u$ and $尾 = \sin^{- 1} v$

So,

$\cos 伪 = u$ where $0 鈮� 伪鈮� 蟺$ and $- 1 鈮� u 鈮� 1$

$\sin 尾 = v$ where $- \frac{蟺}{2} 鈮� 尾鈮� \frac{蟺}{2}$ and $- 1 鈮� v 鈮� 1$

role="math" localid="1646564490741" $\cos 尾 = \sqrt{1 - v^{2}}$

$\sin 伪 = \sqrt{1 - u^{2}}$

Now,

$\cos (伪 + 尾) = \cos 伪路 \cos 尾 - \sin 伪路 \sin 尾 = u \sqrt{1 - v^{2}} - v \sqrt{1 - u^{2}}$

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Most popular questions from this chapter

Show that $\tan^{- 1} (\frac{1}{v}) = \frac{蟺}{2} - \tan^{- 1} v$ , if $v > 0$

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

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

if $\sin 伪 = - \frac{4}{5}$ , $蟺 < 伪 < \frac{3 蟺}{2}$ , then $\cos 伪 =$ _____

Suppose that f and g are two functions with the same domain. If $f (x) = g (x)$ for every x in the domain, the equation is called a(n) ____. Otherwise, it is called a(n) _____ equation.

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