Trigonometry Examples

$\sin (\arccos (\frac{3}{2}))$

Step 1

Draw a triangle in the plane with vertices $(\frac{3}{2}, \sqrt{1^{2} - {(\frac{3}{2})}^{2}})$ , $(\frac{3}{2}, 0)$ , and the origin. Then $\arccos (\frac{3}{2})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\frac{3}{2}, \sqrt{1^{2} - {(\frac{3}{2})}^{2}})$ . Therefore, $\sin (\arccos (\frac{3}{2}))$ is $\sqrt{\frac{- 5}{4}}$ .

$\sqrt{\frac{- 5}{4}}$

Step 2

Move the negative in front of the fraction.

$\sqrt{- \frac{5}{4}}$

Step 3

Rewrite $- \frac{5}{4}$ as ${(\frac{i}{2})}^{2} \cdot 5$ .

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Step 3.1

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

$\sqrt{i^{2} \frac{5}{4}}$

Step 3.2

Factor the perfect power $1^{2}$ out of $5$ .

$\sqrt{i^{2} \frac{1^{2} \cdot 5}{4}}$

Step 3.3

Factor the perfect power $2^{2}$ out of $4$ .

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

Step 3.4

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

$\sqrt{i^{2} ({(\frac{1}{2})}^{2} \cdot 5)}$

Step 3.5

Rewrite $i^{2} {(\frac{1}{2})}^{2}$ as ${(\frac{i}{2})}^{2}$ .

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

Step 4

Pull terms out from under the radical.

$\frac{i}{2} \sqrt{5}$

Step 5

Combine $\frac{i}{2}$ and $\sqrt{5}$ .

$\frac{i \sqrt{5}}{2}$