Trigonometry Examples

$\tan (- 300)$

Step 1

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

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

Step 2

Apply the tangent half-angle identity.

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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.

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

Step 4.3

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

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

Step 4.4

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

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Step 4.4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.4.2

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

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

Step 4.5

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

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

Step 4.6

Combine the numerators over the common denominator.

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

Step 4.7

Add $2$ and $1$ .

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

Step 4.8

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

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

Step 4.9

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.

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

Step 4.10

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

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

Step 4.11

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

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

Step 4.12

Combine the numerators over the common denominator.

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

Step 4.13

Subtract $1$ from $2$ .

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

Step 4.14

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.15

Cancel the common factor of $2$ .

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Step 4.15.1

Cancel the common factor.

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

Step 4.15.2

Rewrite the expression.

$\sqrt{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\sqrt{3}$

Decimal Form:

$1.73205080 \dots$