Precalculus Examples

$\cos (\arcsin (\frac{4}{9}))$

Step 1

Draw a triangle in the plane with vertices $(\sqrt{1^{2} - {(\frac{4}{9})}^{2}}, \frac{4}{9})$ , $(\sqrt{1^{2} - {(\frac{4}{9})}^{2}}, 0)$ , and the origin. Then $\arcsin (\frac{4}{9})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - {(\frac{4}{9})}^{2}}, \frac{4}{9})$ . Therefore, $\cos (\arcsin (\frac{4}{9}))$ is $\sqrt{\frac{65}{81}}$ .

$\sqrt{\frac{65}{81}}$

Step 2

Rewrite $\sqrt{\frac{65}{81}}$ as $\frac{\sqrt{65}}{\sqrt{81}}$ .

$\frac{\sqrt{65}}{\sqrt{81}}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $81$ as $9^{2}$ .

$\frac{\sqrt{65}}{\sqrt{9^{2}}}$

Step 3.2

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

$\frac{\sqrt{65}}{9}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{65}}{9}$

Decimal Form:

$0.89580641 \dots$