Chapter 7: Q. 74 (page 486)

Find the exact value of each expression.
$\sin (\sin^{- 1} \frac{\sqrt{3}}{2} + \cos^{- 1} 1)$

Short Answer

Expert verified

The value of the expression is $\frac{\sqrt{3}}{2}$

Step by step solution

Step 1. Given information

The given expression is $\sin (\sin^{- 1} \frac{\sqrt{3}}{2} + \cos^{- 1} 1)$

Step 2. Solve the expression

Let $α = \sin^{- 1} \frac{\sqrt{3}}{2}$ and $β = \cos^{- 1} 1$

So,

$\sin α = \frac{\sqrt{3}}{2}, - \frac{π}{2} \leq α \leq \frac{π}{2}$ and $\cos β = 1, 0 \leq β \leq π$

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

$\sin β = \sqrt{1 - \cos^{2} β} \sin β = \sqrt{1 - {(1)}^{2}} \sin β = 0$

Now,

$\sin (α + β) = \sin α \cdot \cos β + \cos α \cdot \sin β = \frac{\sqrt{3}}{2} \cdot 1 + \frac{1}{2} \cdot 0 = \frac{\sqrt{3}}{2}$

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Find the exact value of each expression.
$\sin (\sin^{- 1} \frac{\sqrt{3}}{2} + \cos^{- 1} 1)$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Solve the expression

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