Chapter 7: Q. 19 (page 505)

use sum, difference, product, or half-angle formulas to find the exact value of each expression
$\sin (\frac{1}{2} \cos^{- 1} \frac{3}{5})$

Short Answer

Expert verified

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

Step by step solution

Step 1.Given expression

$\sin (\frac{1}{2} \cos^{- 1} \frac{3}{5})$

Step 2. Solving

let

$\frac{1}{2} \cos^{- 1} \frac{3}{5} = θ$

So , we need to calculate for

$\sin θ = ?$

$\frac{1}{2} \cos^{- 1} \frac{3}{5} = θ \cos^{- 1} \frac{3}{5} = 2 θ - 1 ⩽ 2 θ ⩽ 1$

Step 3. Solving for sinθ

$\cos^{- 1} \frac{3}{5} = 2 θ \cos 2 θ = \frac{3}{5}$

We know :

role="math" localid="1646555157092" $1 - 2 \sin^{2} x = \cos 2 x$

Using this :

$\cos 2 θ = \frac{3}{5} 1 - 2 \sin^{2} θ = \frac{3}{5} 1 - \frac{3}{5} = 2 \sin^{2} θ \frac{2}{5} = 2 \sin^{2} θ \sin θ = \frac{1}{\sqrt{5}}$

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use sum, difference, product, or half-angle formulas to find the exact value of each expression
$\sin (\frac{1}{2} \cos^{- 1} \frac{3}{5})$

Short Answer

Step by step solution

Step 1.Given expression

Step 2. Solving

Step 3. Solving for sinθ

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use sum, difference, product, or half-angle formulas to find the exact value of each expressionsin(12cos-135)

Short Answer

Step by step solution

Step 1.Given expression

Step 2. Solving

Step 3. Solving for sinθ

One App. One Place for Learning.

Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

use sum, difference, product, or half-angle formulas to find the exact value of each expression
$\sin (\frac{1}{2} \cos^{- 1} \frac{3}{5})$