Trigonometry Examples

$\cos (\arcsin (\frac{14}{5}))$

Step 1

Draw a triangle in the plane with vertices $(\sqrt{1^{2} - {(\frac{14}{5})}^{2}}, \frac{14}{5})$ , $(\sqrt{1^{2} - {(\frac{14}{5})}^{2}}, 0)$ , and the origin. Then $\arcsin (\frac{14}{5})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - {(\frac{14}{5})}^{2}}, \frac{14}{5})$ . Therefore, $\cos (\arcsin (\frac{14}{5}))$ is $\sqrt{\frac{- 171}{25}}$ .

$\sqrt{\frac{- 171}{25}}$

Step 2

Move the negative in front of the fraction.

$\sqrt{- \frac{171}{25}}$

Step 3

Rewrite $- \frac{171}{25}$ as ${(\frac{3 i}{5})}^{2} \cdot 19$ .

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Step 3.1

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

$\sqrt{i^{2} \frac{171}{25}}$

Step 3.2

Factor the perfect power $3^{2}$ out of $171$ .

$\sqrt{i^{2} \frac{3^{2} \cdot 19}{25}}$

Step 3.3

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

$\sqrt{i^{2} \frac{3^{2} \cdot 19}{5^{2} \cdot 1}}$

Step 3.4

Rearrange the fraction $\frac{3^{2} \cdot 19}{5^{2} \cdot 1}$ .

$\sqrt{i^{2} ({(\frac{3}{5})}^{2} \cdot 19)}$

Step 3.5

Rewrite $i^{2} {(\frac{3}{5})}^{2}$ as ${(\frac{3 i}{5})}^{2}$ .

$\sqrt{{(\frac{3 i}{5})}^{2} \cdot 19}$

Step 4

Pull terms out from under the radical.

$\frac{3 i}{5} \sqrt{19}$

Step 5

Combine $\frac{3 i}{5}$ and $\sqrt{19}$ .

$\frac{3 i \sqrt{19}}{5}$