Trigonometry Examples

$\tan (210) = \frac{\sin (120)}{1 + \cos (60)}$

Step 1

Simplify $\tan (210)$ .

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Step 1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\tan (30) = \frac{\sin (120)}{1 + \cos (60)}$

Step 1.2

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\frac{\sqrt{3}}{3} = \frac{\sin (120)}{1 + \cos (60)}$

Step 2

Simplify $\frac{\sin (120)}{1 + \cos (60)}$ .

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Step 2.1

Simplify the numerator.

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Step 2.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{\sqrt{3}}{3} = \frac{\sin (60)}{1 + \cos (60)}$

Step 2.1.2

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

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

Step 2.2

Simplify the denominator.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Combine the numerators over the common denominator.

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

Step 2.2.4

Add $2$ and $1$ .

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

Step 2.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.4

Cancel the common factor of $2$ .

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Step 2.4.1

Cancel the common factor.

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

Step 2.4.2

Rewrite the expression.

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

Step 2.5

Combine $\sqrt{3}$ and $\frac{1}{3}$ .

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

Step 3

Since $\frac{\sqrt{3}}{3} = \frac{\sqrt{3}}{3}$ , the equation will always be true.

Always true

Step 4

The result can be shown in multiple forms.

Always true

Interval Notation:

$(- \infty, \infty)$