Find the Exact Value cot(arcsin(4/5))

$\cot (\arcsin (\frac{4}{5}))$

Draw a triangle in the plane with vertices $(\sqrt{1^{2} - {(\frac{4}{5})}^{2}}, \frac{4}{5})$ , $(\sqrt{1^{2} - {(\frac{4}{5})}^{2}}, 0)$ , and the origin. Then $\arcsin (\frac{4}{5})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - {(\frac{4}{5})}^{2}}, \frac{4}{5})$ . Therefore, $\cot (\arcsin (\frac{4}{5}))$ is $\frac{\sqrt{1^{2} - {(\frac{4}{5})}^{2}}}{\frac{4}{5}}$ .

$\frac{\sqrt{1^{2} - {(\frac{4}{5})}^{2}}}{\frac{4}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{1^{2} - {(\frac{4}{5})}^{2}} \frac{5}{4}$

One to any power is one.

$\sqrt{1 - {(\frac{4}{5})}^{2}} \frac{5}{4}$

Apply the product rule to $\frac{4}{5}$ .

$\sqrt{1 - \frac{4^{2}}{5^{2}}} \frac{5}{4}$

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

$\sqrt{1 - \frac{16}{5^{2}}} \frac{5}{4}$

Raise $5$ to the power of $2$ .

$\sqrt{1 - \frac{16}{25}} \frac{5}{4}$

To write $\frac{1}{1}$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$\sqrt{\frac{1}{1} \cdot \frac{25}{25} - \frac{16}{25}} \frac{5}{4}$

Write each expression with a common denominator of $25$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$\sqrt{\frac{1 \cdot 25}{1 \cdot 25} - \frac{16}{25}} \frac{5}{4}$

Multiply $25$ by $1$ .

$\sqrt{\frac{1 \cdot 25}{25} - \frac{16}{25}} \frac{5}{4}$

Combine the numerators over the common denominator.

$\sqrt{\frac{1 \cdot 25 - 16}{25}} \frac{5}{4}$

Simplify the numerator.

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Multiply $1$ by $25$ .

$\sqrt{\frac{25 - 16}{25}} \frac{5}{4}$

Subtract $16$ from $25$ .

$\sqrt{\frac{9}{25}} \frac{5}{4}$

Rewrite $\sqrt{\frac{9}{25}}$ as $\frac{\sqrt{9}}{\sqrt{25}}$ .

$\frac{\sqrt{9}}{\sqrt{25}} \cdot \frac{5}{4}$

Simplify the numerator.

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Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2}}}{\sqrt{25}} \cdot \frac{5}{4}$

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

$\frac{3}{\sqrt{25}} \cdot \frac{5}{4}$

Simplify the denominator.

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Rewrite $25$ as $5^{2}$ .

$\frac{3}{\sqrt{5^{2}}} \cdot \frac{5}{4}$

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

$\frac{3}{5} \cdot \frac{5}{4}$

Simplify terms.

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Cancel the common factor of $5$ .

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Cancel the common factor.

$\frac{3}{5} \cdot \frac{5}{4}$

Rewrite the expression.

$3 (\frac{1}{4})$

Combine $3$ and $\frac{1}{4}$ .

$\frac{3}{4}$

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{4}$

Decimal Form:

$0.75$

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