Evaluate cot(-300 degrees )

$\cot (- 300 °)$

Rewrite $- 300 °$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\cot (\frac{- 600}{2})$

Apply the reciprocal identity.

$\frac{1}{\tan (\frac{- 600}{2})}$

Apply the tangent half-angle identity.

$\frac{1}{\pm \sqrt{\frac{1 - \cos (- 600)}{1 + \cos (- 600)}}}$

Change the $\pm$ to $+$ because cotangent is positive in the first quadrant.

$\frac{1}{\sqrt{\frac{1 - \cos (- 600)}{1 + \cos (- 600)}}}$

Simplify $\frac{1}{\sqrt{\frac{1 - \cos (- 600)}{1 + \cos (- 600)}}}$ .

Simplify the numerator.

Add full rotations of $360$ ° until the angle is between $0$ ° and $360$ °.

$\frac{1}{\sqrt{\frac{1 - \cos (120)}{1 + \cos (- 600)}}}$

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$\frac{1}{\sqrt{\frac{1 - - \cos (60)}{1 + \cos (- 600)}}}$

The exact value of $\cos (60)$ is $\frac{1}{2}$ .

$\frac{1}{\sqrt{\frac{1 - - \frac{1}{2}}{1 + \cos (- 600)}}}$

Multiply $- - \frac{1}{2}$ .

Multiply $- 1$ by $- 1$ .

$\frac{1}{\sqrt{\frac{1 + 1 (\frac{1}{2})}{1 + \cos (- 600)}}}$

Multiply $\frac{1}{2}$ by $1$ .

$\frac{1}{\sqrt{\frac{1 + \frac{1}{2}}{1 + \cos (- 600)}}}$

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

$\frac{1}{\sqrt{\frac{\frac{2}{2} + \frac{1}{2}}{1 + \cos (- 600)}}}$

Combine the numerators over the common denominator.

$\frac{1}{\sqrt{\frac{\frac{2 + 1}{2}}{1 + \cos (- 600)}}}$

Add $2$ and $1$ .

$\frac{1}{\sqrt{\frac{\frac{3}{2}}{1 + \cos (- 600)}}}$

Simplify the denominator.

Add full rotations of $360$ ° until the angle is between $0$ ° and $360$ °.

$\frac{1}{\sqrt{\frac{\frac{3}{2}}{1 + \cos (120)}}}$

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$\frac{1}{\sqrt{\frac{\frac{3}{2}}{1 - \cos (60)}}}$

The exact value of $\cos (60)$ is $\frac{1}{2}$ .

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

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

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

Combine the numerators over the common denominator.

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

Subtract $1$ from $2$ .

$\frac{1}{\sqrt{\frac{\frac{3}{2}}{\frac{1}{2}}}}$

Simplify the denominator.

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{\sqrt{\frac{3}{2} \cdot 2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{1}{\sqrt{\frac{3}{2} \cdot 2}}$

Rewrite the expression.

$\frac{1}{\sqrt{3}}$

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

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

$\frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

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

$\frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{\sqrt{3}}{3^{1}}$

Evaluate the exponent.

$\frac{\sqrt{3}}{3}$

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{3}}{3}$

Decimal Form:

$0.57735026 \dots$

Evaluate cot(-300 degrees )

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