Chapter 7: Q. 75 (page 486)

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

Short Answer

Expert verified

The value of the expression is $- \frac{24}{25}$

Step by step solution

Step 1. Given information

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

Step 2. Solve the expression

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

So,

$\sin α = \frac{3}{5}, - \frac{π}{2} \leq α \leq \frac{π}{2}$ and $\cos β = - \frac{4}{5}, 0 \leq β \leq π$

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

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

Now,

$\sin (α - β) = \sin α \cdot \cos β - \cos α \cdot \sin β = \frac{3}{5} \cdot (- \frac{4}{5}) - \frac{4}{5} \cdot \frac{3}{5} = - \frac{24}{25}$

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

Short Answer

Step by step solution

Step 1. Given information

Step 2. Solve the expression

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