Precalculus Examples

$y = \cos (\arcsin (\frac{1}{6}))$

Step 1

Draw a triangle in the plane with vertices $(\sqrt{1^{2} - {(\frac{1}{6})}^{2}}, \frac{1}{6})$ , $(\sqrt{1^{2} - {(\frac{1}{6})}^{2}}, 0)$ , and the origin. Then $\arcsin (\frac{1}{6})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - {(\frac{1}{6})}^{2}}, \frac{1}{6})$ . Therefore, $\cos (\arcsin (\frac{1}{6}))$ is $\sqrt{\frac{35}{36}}$ .

$y = \sqrt{\frac{35}{36}}$

Step 2

Rewrite $\sqrt{\frac{35}{36}}$ as $\frac{\sqrt{35}}{\sqrt{36}}$ .

$y = \frac{\sqrt{35}}{\sqrt{36}}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $36$ as $6^{2}$ .

$y = \frac{\sqrt{35}}{\sqrt{6^{2}}}$

Step 3.2

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

$y = \frac{\sqrt{35}}{6}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$y = \frac{\sqrt{35}}{6}$

Decimal Form:

$y = 0.98601329 \dots$