Trigonometry Examples

$\sqrt{16 - {(4 \sin (x))}^{2}}$

Step 1

Rewrite $16$ as $4^{2}$ .

$\sqrt{4^{2} - {(4 \sin (x))}^{2}}$

Step 2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4$ and $b = 4 \sin (x)$ .

$\sqrt{(4 + 4 \sin (x)) (4 - (4 \sin (x)))}$

Step 3

Simplify.

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Step 3.1

Factor $4$ out of $4 + 4 \sin (x)$ .

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Step 3.1.1

Factor $4$ out of $4$ .

$\sqrt{(4 \cdot 1 + 4 \sin (x)) (4 - (4 \sin (x)))}$

Step 3.1.2

Factor $4$ out of $4 \cdot 1 + 4 \sin (x)$ .

$\sqrt{4 (1 + \sin (x)) (4 - (4 \sin (x)))}$

Step 3.2

Multiply $4$ by $- 1$ .

$\sqrt{4 (1 + \sin (x)) (4 - 4 \sin (x))}$

Step 4

Factor $4$ out of $4 - 4 \sin (x)$ .

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Step 4.1

Factor $4$ out of $4$ .

$\sqrt{4 (1 + \sin (x)) (4 (1) - 4 \sin (x))}$

Step 4.2

Factor $4$ out of $- 4 \sin (x)$ .

$\sqrt{4 (1 + \sin (x)) (4 (1) + 4 (- \sin (x)))}$

Step 4.3

Factor $4$ out of $4 (1) + 4 (- \sin (x))$ .

$\sqrt{4 (1 + \sin (x)) (4 (1 - \sin (x)))}$

$\sqrt{4 (1 + \sin (x)) \cdot 4 (1 - \sin (x))}$

Step 5

Multiply $4$ by $4$ .

$\sqrt{16 (1 + \sin (x)) (1 - \sin (x))}$

Step 6

Rewrite $16 (1 + \sin (x)) (1 - \sin (x))$ as ${(2^{2})}^{2} ((1 + \sin (x)) (1 - \sin (x)))$ .

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Step 6.1

Rewrite $16$ as $4^{2}$ .

$\sqrt{4^{2} (1 + \sin (x)) (1 - \sin (x))}$

Step 6.2

Rewrite $4$ as $2^{2}$ .

$\sqrt{{(2^{2})}^{2} (1 + \sin (x)) (1 - \sin (x))}$

Step 6.3

Add parentheses.

$\sqrt{{(2^{2})}^{2} ((1 + \sin (x)) (1 - \sin (x)))}$

Step 7

Pull terms out from under the radical.

$2^{2} \sqrt{(1 + \sin (x)) (1 - \sin (x))}$

Step 8

Raise $2$ to the power of $2$ .

$4 \sqrt{(1 + \sin (x)) (1 - \sin (x))}$