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 $- 1 \leq u \leq 1, - 1 \leq v \leq 1$

Step by step solution

Step 1. Given information

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

Step 2. Solve the expression

Let $α = \cos^{- 1} u$ and $β = \sin^{- 1} v$

So,

$\cos α = u$ where $0 \leq α \leq π$ and $- 1 \leq u \leq 1$

$\sin β = v$ where $- \frac{π}{2} \leq β \leq \frac{π}{2}$ and $- 1 \leq v \leq 1$

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

$\sin α = \sqrt{1 - u^{2}}$

Now,

$\cos (α + β) = \cos α \cdot \cos β - \sin α \cdot \sin β = u \sqrt{1 - v^{2}} - v \sqrt{1 - u^{2}}$

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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

Step by step solution

Step 1. Given information

Step 2. Solve the expression

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Write each trigonometric expression as an algebraic expression containing uand v. Give the restrictions required on uand v.role="math" localid="1646564067411" coscos-1u+sin-1v

Short Answer

Step by step solution

Step 1. Given information

Step 2. Solve the expression

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Probability and Statistics

Statistics

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Study anywhere. Anytime. Across all devices.

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