Chapter 7: Q. 76 (page 486)

Find the exact value of each expression.
$\sin (\sin^{- 1} (- \frac{4}{5}) - \tan^{- 1} \frac{3}{4})$

Short Answer

Expert verified

The value of the expression is $- 1$

Step by step solution

Step 1. Given information

The given expression is $\sin (\sin^{- 1} (- \frac{4}{5}) - \tan^{- 1} \frac{3}{4})$

Step 2. Solve the expression

Let $α = \sin^{- 1} (- \frac{4}{5})$ and $β = \tan^{- 1} \frac{3}{4}$

So,

role="math" localid="1646553906966" $\sin α = - \frac{4}{5}, - \frac{π}{2} \leq α \leq \frac{π}{2}$

$\tan β = \frac{3}{4}, - \frac{π}{2} \leq β \leq \frac{π}{2}$

$\cos α = \sqrt{1 - \sin^{2} α} \cos α = \sqrt{1 - {(- \frac{4}{5})}^{2}} \cos α = \frac{3}{5}$

$s e c^{2} β = 1 + \tan^{2} β s e c^{2} β = 1 + {(\frac{3}{4})}^{2} s e c^{2} β = = \frac{25}{16} s e c β = \frac{5}{4} \cos β = \frac{4}{5}$

$\sin β = \sqrt{1 - \cos^{2} β} \sin β = \sqrt{1 - {(\frac{4}{5})}^{2}} \sin β = \frac{3}{5}$

Now,

$\sin (α - β) = \sin α \cdot \cos β - \cos α \cdot \sin β = - \frac{4}{5} \cdot \frac{4}{5} - \frac{3}{5} \cdot \frac{3}{5} = - 1$

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Find the exact value of each expression.
$\sin (\sin^{- 1} (- \frac{4}{5}) - \tan^{- 1} \frac{3}{4})$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Solve the expression

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Step by step solution

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Most popular questions from this chapter

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Find the exact value of each expression.
$\sin (\sin^{- 1} (- \frac{4}{5}) - \tan^{- 1} \frac{3}{4})$