Trigonometry Examples

$\cot (arccsc (\sqrt{10}))$

Step 1

Draw a triangle in the plane with vertices $(\sqrt{{\sqrt{10}}^{2} - 1^{2}}, 1)$ , $(\sqrt{{\sqrt{10}}^{2} - 1^{2}}, 0)$ , and the origin. Then $arccsc (\sqrt{10})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{{\sqrt{10}}^{2} - 1^{2}}, 1)$ . Therefore, $\cot (arccsc (\sqrt{10}))$ is $\sqrt{{\sqrt{10}}^{2} - 1}$ .

$\sqrt{{\sqrt{10}}^{2} - 1}$

Step 2

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\sqrt{{(10^{\frac{1}{2}})}^{2} - 1}$

Step 2.2

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

$\sqrt{10^{\frac{1}{2} \cdot 2} - 1}$

Step 2.3

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

$\sqrt{10^{\frac{2}{2}} - 1}$

Step 2.4

Cancel the common factor of $2$ .

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Step 2.4.1

Cancel the common factor.

$\sqrt{10^{\frac{2}{2}} - 1}$

Step 2.4.2

Rewrite the expression.

$\sqrt{10^{1} - 1}$

Step 2.5

Evaluate the exponent.

$\sqrt{10 - 1}$

Step 3

Simplify the expression.

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Step 3.1

Subtract $1$ from $10$ .

$\sqrt{9}$

Step 3.2

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

$\sqrt{3^{2}}$

Step 4

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

$3$