Trigonometry Examples

$\tan (- 330)$

Step 1

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

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

Step 2

Apply the tangent half-angle identity.

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Combine the numerators over the common denominator.

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

Step 4.5

Subtract $1$ from $2$ .

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

Step 4.6

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

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

Step 4.7

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

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

Step 4.8

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

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

Step 4.9

Combine the numerators over the common denominator.

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

Step 4.10

Add $2$ and $1$ .

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

Step 4.11

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.12

Cancel the common factor of $2$ .

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Step 4.12.1

Cancel the common factor.

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

Step 4.12.2

Rewrite the expression.

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

Step 4.13

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

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

Step 4.14

Any root of $1$ is $1$ .

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

Step 4.15

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

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

Step 4.16

Combine and simplify the denominator.

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Step 4.16.1

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

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

Step 4.16.2

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

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

Step 4.16.3

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

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

Step 4.16.4

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

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

Step 4.16.5

Add $1$ and $1$ .

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

Step 4.16.6

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

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Step 4.16.6.1

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

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

Step 4.16.6.2

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

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

Step 4.16.6.3

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

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

Step 4.16.6.4

Cancel the common factor of $2$ .

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Step 4.16.6.4.1

Cancel the common factor.

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

Step 4.16.6.4.2

Rewrite the expression.

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

Step 4.16.6.5

Evaluate the exponent.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.57735026 \dots$