Chapter 7: Q. 79 (page 486)

Find the exact value of each expression.
$\cos (\sin^{- 1} \frac{5}{13} - \tan^{- 1} \frac{3}{4})$

Short Answer

Expert verified

The value of the expression is $\frac{63}{65}$

Step by step solution

Step 1. Given information

The given expression is $\cos (\sin^{- 1} \frac{5}{13} - \tan^{- 1} \frac{3}{4})$

Step 2. Solve the expression

Let $α = \sin^{- 1} \frac{5}{13}$ and $β = \tan^{- 1} \frac{3}{4}$

So,

$\sin α = \frac{5}{13}, - \frac{π}{2} \leq α \leq \frac{π}{2}$

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

$\cos α = \frac{12}{13}$

$\sin β = \frac{3}{5} \cos β = \frac{4}{5}$

Now,

$\cos (α - β) = \cos α \cdot \cos β + \sin α \cdot \sin β = \frac{12}{13} \cdot \frac{4}{5} + \frac{5}{13} \cdot \frac{3}{5} = \frac{63}{65}$

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Find the exact value of each expression.
$\cos (\sin^{- 1} \frac{5}{13} - \tan^{- 1} \frac{3}{4})$

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.
$\cos (\sin^{- 1} \frac{5}{13} - \tan^{- 1} \frac{3}{4})$