Trigonometry Examples

$\tan (- 210)$

Step 1

The angle $- 210$ is an angle where the values of the six trigonometric functions are known. Because this is the case, add $0$ to keep the value the same.

$\tan (- 210 + 0)$

Step 2

Use the difference formula for tangent to simplify the expression. The formula states that $\tan (A - B) = \frac{\tan (A) - \tan (B)}{1 + \tan (A) \tan (B)}$ .

$\frac{\tan (0) - \tan (210)}{1 + \tan (0) \tan (210)}$

Step 3

Remove parentheses.

$\frac{\tan (0) - \tan (210)}{1 + \tan (0) \tan (210)}$

Step 4

Simplify the numerator.

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Step 4.1

The exact value of $\tan (0)$ is $0$ .

$\frac{0 - \tan (210)}{1 + \tan (0) \tan (210)}$

Step 4.2

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

$\frac{0 - \tan (30)}{1 + \tan (0) \tan (210)}$

Step 4.3

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

$\frac{0 - \frac{\sqrt{3}}{3}}{1 + \tan (0) \tan (210)}$

Step 4.4

Subtract $\frac{\sqrt{3}}{3}$ from $0$ .

$\frac{- \frac{\sqrt{3}}{3}}{1 + \tan (0) \tan (210)}$

Step 5

Simplify the denominator.

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Step 5.1

The exact value of $\tan (0)$ is $0$ .

$\frac{- \frac{\sqrt{3}}{3}}{1 + 0 \tan (210)}$

Step 5.2

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

$\frac{- \frac{\sqrt{3}}{3}}{1 + 0 \tan (30)}$

Step 5.3

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

$\frac{- \frac{\sqrt{3}}{3}}{1 + 0 \frac{\sqrt{3}}{3}}$

Step 5.4

Multiply $0$ by $\frac{\sqrt{3}}{3}$ .

$\frac{- \frac{\sqrt{3}}{3}}{1 + 0}$

Step 5.5

Add $1$ and $0$ .

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

Step 6

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

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.57735026 \dots$