Precalculus Examples

$\sin (\arccos (\frac{21}{29}))$

Step 1

Draw a triangle in the plane with vertices $(\frac{21}{29}, \sqrt{1^{2} - {(\frac{21}{29})}^{2}})$ , $(\frac{21}{29}, 0)$ , and the origin. Then $\arccos (\frac{21}{29})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\frac{21}{29}, \sqrt{1^{2} - {(\frac{21}{29})}^{2}})$ . Therefore, $\sin (\arccos (\frac{21}{29}))$ is $\sqrt{\frac{400}{841}}$ .

$\sqrt{\frac{400}{841}}$

Step 2

Rewrite $\sqrt{\frac{400}{841}}$ as $\frac{\sqrt{400}}{\sqrt{841}}$ .

$\frac{\sqrt{400}}{\sqrt{841}}$

Step 3

Simplify the numerator.

Tap for more steps...

Step 3.1

Rewrite $400$ as $20^{2}$ .

$\frac{\sqrt{20^{2}}}{\sqrt{841}}$

Step 3.2

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

$\frac{20}{\sqrt{841}}$

Step 4

Simplify the denominator.

Tap for more steps...

Step 4.1

Rewrite $841$ as $29^{2}$ .

$\frac{20}{\sqrt{29^{2}}}$

Step 4.2

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

$\frac{20}{29}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{20}{29}$

Decimal Form:

$0.68965517 \dots$