Trigonometry Examples

$\cot (arccsc (- \frac{13}{12}))$

Step 1

Draw a triangle in the plane with vertices $(\sqrt{{(- \frac{13}{12})}^{2} - 1^{2}}, 1)$ , $(\sqrt{{(- \frac{13}{12})}^{2} - 1^{2}}, 0)$ , and the origin. Then $arccsc (- \frac{13}{12})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{{(- \frac{13}{12})}^{2} - 1^{2}}, 1)$ . Therefore, $\cot (arccsc (- \frac{13}{12}))$ is $\sqrt{{(- \frac{13}{12})}^{2} - 1}$ .

$\sqrt{{(- \frac{13}{12})}^{2} - 1}$

Step 2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 2.1

Apply the product rule to $- \frac{13}{12}$ .

$\sqrt{{(- 1)}^{2} {(\frac{13}{12})}^{2} - 1}$

Step 2.2

Apply the product rule to $\frac{13}{12}$ .

$\sqrt{{(- 1)}^{2} \frac{13^{2}}{12^{2}} - 1}$

Step 3

Raise $- 1$ to the power of $2$ .

$\sqrt{1 \frac{13^{2}}{12^{2}} - 1}$

Step 4

Multiply $\frac{13^{2}}{12^{2}}$ by $1$ .

$\sqrt{\frac{13^{2}}{12^{2}} - 1}$

Step 5

Raise $13$ to the power of $2$ .

$\sqrt{\frac{169}{12^{2}} - 1}$

Step 6

Raise $12$ to the power of $2$ .

$\sqrt{\frac{169}{144} - 1}$

Step 7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{144}{144}$ .

$\sqrt{\frac{169}{144} - 1 \cdot \frac{144}{144}}$

Step 8

Combine $- 1$ and $\frac{144}{144}$ .

$\sqrt{\frac{169}{144} + \frac{- 1 \cdot 144}{144}}$

Step 9

Combine the numerators over the common denominator.

$\sqrt{\frac{169 - 1 \cdot 144}{144}}$

Step 10

Simplify the numerator.

Tap for more steps...

Step 10.1

Multiply $- 1$ by $144$ .

$\sqrt{\frac{169 - 144}{144}}$

Step 10.2

Subtract $144$ from $169$ .

$\sqrt{\frac{25}{144}}$

Step 11

Rewrite $\sqrt{\frac{25}{144}}$ as $\frac{\sqrt{25}}{\sqrt{144}}$ .

$\frac{\sqrt{25}}{\sqrt{144}}$

Step 12

Simplify the numerator.

Tap for more steps...

Step 12.1

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

$\frac{\sqrt{5^{2}}}{\sqrt{144}}$

Step 12.2

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

$\frac{5}{\sqrt{144}}$

Step 13

Simplify the denominator.

Tap for more steps...

Step 13.1

Rewrite $144$ as $12^{2}$ .

$\frac{5}{\sqrt{12^{2}}}$

Step 13.2

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

$\frac{5}{12}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{12}$

Decimal Form:

$0.41\overline{6}$