Precalculus Examples

$\cos^{2} (\arcsin (\frac{1}{4}))$

Step 1

Draw a triangle in the plane with vertices $(\sqrt{1^{2} - {(\frac{1}{4})}^{2}}, \frac{1}{4})$ , $(\sqrt{1^{2} - {(\frac{1}{4})}^{2}}, 0)$ , and the origin. Then $\arcsin (\frac{1}{4})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - {(\frac{1}{4})}^{2}}, \frac{1}{4})$ . Therefore, $\cos (\arcsin (\frac{1}{4}))$ is $\sqrt{\frac{15}{16}}$ .

${\sqrt{\frac{15}{16}}}^{2}$

Step 2

Rewrite $\sqrt{\frac{15}{16}}$ as $\frac{\sqrt{15}}{\sqrt{16}}$ .

${(\frac{\sqrt{15}}{\sqrt{16}})}^{2}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $16$ as $4^{2}$ .

${(\frac{\sqrt{15}}{\sqrt{4^{2}}})}^{2}$

Step 3.2

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

${(\frac{\sqrt{15}}{4})}^{2}$

Step 4

Simplify by cancelling exponent with radical.

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Step 4.1

Apply the product rule to $\frac{\sqrt{15}}{4}$ .

$\frac{{\sqrt{15}}^{2}}{4^{2}}$

Step 4.2

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$\frac{{(15^{\frac{1}{2}})}^{2}}{4^{2}}$

Step 4.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{15^{\frac{1}{2} \cdot 2}}{4^{2}}$

Step 4.2.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{15^{\frac{2}{2}}}{4^{2}}$

Step 4.2.4

Cancel the common factor of $2$ .

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Step 4.2.4.1

Cancel the common factor.

$\frac{15^{\frac{2}{2}}}{4^{2}}$

Step 4.2.4.2

Rewrite the expression.

$\frac{15^{1}}{4^{2}}$

Step 4.2.5

Evaluate the exponent.

$\frac{15}{4^{2}}$

Step 4.3

Raise $4$ to the power of $2$ .

$\frac{15}{16}$