Trigonometry Examples

$\sin (\arccos (\frac{1}{2}))$

Step 1

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

$\sqrt{\frac{3}{4}}$

Step 2

Rewrite $\sqrt{\frac{3}{4}}$ as $\frac{\sqrt{3}}{\sqrt{4}}$ .

$\frac{\sqrt{3}}{\sqrt{4}}$

Step 3

Simplify the denominator.

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Step 3.1

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

$\frac{\sqrt{3}}{\sqrt{2^{2}}}$

Step 3.2

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

$\frac{\sqrt{3}}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{3}}{2}$

Decimal Form:

$0.86602540 \dots$