Trigonometry Examples

$\cos (\arcsin (\frac{21}{29}))$

Step 1

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

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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.

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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:

$0.68965517 \dots$