Find the Exact Value cot(arcsin(3/5))

$\cot (\arcsin (\frac{3}{5}))$

Draw a triangle in the plane with vertices $(\sqrt{1^{2} - {(\frac{3}{5})}^{2}}, \frac{3}{5})$ , $(\sqrt{1^{2} - {(\frac{3}{5})}^{2}}, 0)$ , and the origin. Then $\arcsin (\frac{3}{5})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - {(\frac{3}{5})}^{2}}, \frac{3}{5})$ . Therefore, $\cot (\arcsin (\frac{3}{5}))$ is $\frac{\sqrt{1^{2} - {(\frac{3}{5})}^{2}}}{\frac{3}{5}}$ .

$\frac{\sqrt{1^{2} - {(\frac{3}{5})}^{2}}}{\frac{3}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{1^{2} - {(\frac{3}{5})}^{2}} \frac{5}{3}$

One to any power is one.

$\sqrt{1 - {(\frac{3}{5})}^{2}} \frac{5}{3}$

Apply the product rule to $\frac{3}{5}$ .

$\sqrt{1 - \frac{3^{2}}{5^{2}}} \frac{5}{3}$

Raise $3$ to the power of $2$ .

$\sqrt{1 - \frac{9}{5^{2}}} \frac{5}{3}$

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

$\sqrt{1 - \frac{9}{25}} \frac{5}{3}$

Write $1$ as a fraction with a common denominator.

$\sqrt{\frac{25}{25} - \frac{9}{25}} \frac{5}{3}$

Combine the numerators over the common denominator.

$\sqrt{\frac{25 - 9}{25}} \frac{5}{3}$

Subtract $9$ from $25$ .

$\sqrt{\frac{16}{25}} \frac{5}{3}$

Rewrite $\sqrt{\frac{16}{25}}$ as $\frac{\sqrt{16}}{\sqrt{25}}$ .

$\frac{\sqrt{16}}{\sqrt{25}} \cdot \frac{5}{3}$

Simplify the numerator.

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

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

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

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

Simplify the denominator.

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

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

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

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

Simplify terms.

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Cancel the common factor of $5$ .

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Cancel the common factor.

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

Rewrite the expression.

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

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

$\frac{4}{3}$

The result can be shown in multiple forms.

Exact Form:

$\frac{4}{3}$

Decimal Form:

$1.\overline{3}$

Mixed Number Form:

$1 \frac{1}{3}$

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