Trigonometry Examples

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

Step 1

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

$\sqrt{\frac{1}{4}}$

Step 2

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$\frac{\sqrt{1}}{\sqrt{4}}$

Step 3

Any root of $1$ is $1$ .

$\frac{1}{\sqrt{4}}$

Step 4

Simplify the denominator.

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Step 4.1

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

$\frac{1}{\sqrt{2^{2}}}$

Step 4.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{1}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$