Trigonometry Examples

$\cot (- 330)$

Step 1

Rewrite $- 330$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\cot (\frac{- 660}{2})$

Step 2

Apply the reciprocal identity.

$\frac{1}{\tan (\frac{- 660}{2})}$

Step 3

Apply the tangent half-angle identity.

$\frac{1}{\pm \sqrt{\frac{1 - \cos (- 660)}{1 + \cos (- 660)}}}$

Step 4

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

$\frac{1}{\sqrt{\frac{1 - \cos (- 660)}{1 + \cos (- 660)}}}$

Step 5

Simplify $\frac{1}{\sqrt{\frac{1 - \cos (- 660)}{1 + \cos (- 660)}}}$ .

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$\frac{1}{\sqrt{\frac{1 - \cos (60)}{1 + \cos (- 660)}}}$

Step 5.1.2

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

$\frac{1}{\sqrt{\frac{1 - \frac{1}{2}}{1 + \cos (- 660)}}}$

Step 5.1.3

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

$\frac{1}{\sqrt{\frac{\frac{2}{2} - \frac{1}{2}}{1 + \cos (- 660)}}}$

Step 5.1.4

Combine the numerators over the common denominator.

$\frac{1}{\sqrt{\frac{\frac{2 - 1}{2}}{1 + \cos (- 660)}}}$

Step 5.1.5

Subtract $1$ from $2$ .

$\frac{1}{\sqrt{\frac{\frac{1}{2}}{1 + \cos (- 660)}}}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.2.4

Combine the numerators over the common denominator.

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

Step 5.2.5

Add $2$ and $1$ .

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

Step 5.3

Simplify the denominator.

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Step 5.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.2

Cancel the common factor of $2$ .

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Step 5.3.2.1

Cancel the common factor.

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

Step 5.3.2.2

Rewrite the expression.

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

Step 5.3.3

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

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

Step 5.3.4

Any root of $1$ is $1$ .

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

Step 5.3.5

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

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

Step 5.3.6

Combine and simplify the denominator.

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Step 5.3.6.1

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

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

Step 5.3.6.2

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

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

Step 5.3.6.3

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

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

Step 5.3.6.4

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

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

Step 5.3.6.5

Add $1$ and $1$ .

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

Step 5.3.6.6

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

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Step 5.3.6.6.1

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

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

Step 5.3.6.6.2

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

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

Step 5.3.6.6.3

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

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

Step 5.3.6.6.4

Cancel the common factor of $2$ .

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Step 5.3.6.6.4.1

Cancel the common factor.

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

Step 5.3.6.6.4.2

Rewrite the expression.

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

Step 5.3.6.6.5

Evaluate the exponent.

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

Step 5.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.5

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

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

Step 5.6

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

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

Step 5.7

Combine and simplify the denominator.

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Step 5.7.1

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

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

Step 5.7.2

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

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

Step 5.7.3

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

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

Step 5.7.4

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

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

Step 5.7.5

Add $1$ and $1$ .

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

Step 5.7.6

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

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Step 5.7.6.1

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

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

Step 5.7.6.2

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

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

Step 5.7.6.3

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

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

Step 5.7.6.4

Cancel the common factor of $2$ .

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Step 5.7.6.4.1

Cancel the common factor.

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

Step 5.7.6.4.2

Rewrite the expression.

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

Step 5.7.6.5

Evaluate the exponent.

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

Step 5.8

Cancel the common factor of $3$ .

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Step 5.8.1

Cancel the common factor.

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

Step 5.8.2

Divide $\sqrt{3}$ by $1$ .

$\sqrt{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\sqrt{3}$

Decimal Form:

$1.73205080 \dots$