Trigonometry Examples

$\cos (\arcsin (\frac{x}{5}))$

Step 1

Write the expression using exponents.

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Draw a triangle in the plane with vertices $(\sqrt{1^{2} - {(\frac{x}{5})}^{2}}, \frac{x}{5})$ , $(\sqrt{1^{2} - {(\frac{x}{5})}^{2}}, 0)$ , and the origin. Then $\arcsin (\frac{x}{5})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - {(\frac{x}{5})}^{2}}, \frac{x}{5})$ . Therefore, $\cos (\arcsin (\frac{x}{5}))$ is $\sqrt{1 - {(\frac{x}{5})}^{2}}$ .

$\sqrt{1 - {(\frac{x}{5})}^{2}}$

Rewrite $1$ as $1^{2}$ .

$\sqrt{1^{2} - {(\frac{x}{5})}^{2}}$

Step 2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \frac{x}{5}$ .

$\sqrt{(1 + \frac{x}{5}) (1 - \frac{x}{5})}$

Step 3

Simplify terms.

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Write $1$ as a fraction with a common denominator.

$\sqrt{(\frac{5}{5} + \frac{x}{5}) (1 - \frac{x}{5})}$

Combine the numerators over the common denominator.

$\sqrt{\frac{5 + x}{5} (1 - \frac{x}{5})}$

Write $1$ as a fraction with a common denominator.

$\sqrt{\frac{5 + x}{5} (\frac{5}{5} - \frac{x}{5})}$

Combine the numerators over the common denominator.

$\sqrt{\frac{5 + x}{5} \cdot \frac{5 - x}{5}}$

Multiply $\frac{5 + x}{5}$ by $\frac{5 - x}{5}$ .

$\sqrt{\frac{(5 + x) (5 - x)}{5 \cdot 5}}$

Multiply $5$ by $5$ .

$\sqrt{\frac{(5 + x) (5 - x)}{25}}$

Step 4

Rewrite $\frac{(5 + x) (5 - x)}{25}$ as ${(\frac{1}{5})}^{2} ((5 + x) (5 - x))$ .

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Factor the perfect power $1^{2}$ out of $(5 + x) (5 - x)$ .

$\sqrt{\frac{1^{2} ((5 + x) (5 - x))}{25}}$

Factor the perfect power $5^{2}$ out of $25$ .

$\sqrt{\frac{1^{2} ((5 + x) (5 - x))}{5^{2} \cdot 1}}$

Rearrange the fraction $\frac{1^{2} ((5 + x) (5 - x))}{5^{2} \cdot 1}$ .

$\sqrt{{(\frac{1}{5})}^{2} ((5 + x) (5 - x))}$

Step 5

Pull terms out from under the radical.

$\frac{1}{5} \sqrt{(5 + x) (5 - x)}$

Step 6

Combine $\frac{1}{5}$ and $\sqrt{(5 + x) (5 - x)}$ .

$\frac{\sqrt{(5 + x) (5 - x)}}{5}$