Find the Exact Value cot(arccsc(-3))

$\cot (arccsc (- 3))$

Step 1

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

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

Step 2

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

$\sqrt{9 - 1}$

Step 3

Subtract $1$ from $9$ .

$\sqrt{8}$

Step 4

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$\sqrt{4 (2)}$

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

$\sqrt{2^{2} \cdot 2}$

Step 5

Pull terms out from under the radical.

$2 \sqrt{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{2}$

Decimal Form:

$2.82842712 \dots$

Find the Exact Value cot(arccsc(-3))

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