Trigonometry Examples

$\tan (x) = - \frac{\sqrt{3}}{3}$

Step 1

Take the inverse tangent of both sides of the equation to extract $x$ from inside the tangent.

$x = \arctan (- \frac{\sqrt{3}}{3})$

Step 2

Simplify the right side.

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Step 2.1

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

$x = - 30$

Step 3

The tangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the third quadrant.

$x = - 30 - 180$

Step 4

Simplify the expression to find the second solution.

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Step 4.1

Add $360 °$ to $- 30 - 180 °$ .

$x = - 30 - 180 ° + 360 °$

Step 4.2

The resulting angle of $150 °$ is positive and coterminal with $- 30 - 180$ .

$x = 150 °$

Step 5

Find the period of $\tan (x)$ .

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Step 5.1

The period of the function can be calculated using $\frac{180}{| b |}$ .

$\frac{180}{| b |}$

Step 5.2

Replace $b$ with $1$ in the formula for period.

$\frac{180}{| 1 |}$

Step 5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{180}{1}$

Step 5.4

Divide $180$ by $1$ .

$180$

Step 6

Add $180$ to every negative angle to get positive angles.

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Step 6.1

Add $180$ to $- 30$ to find the positive angle.

$- 30 + 180$

Step 6.2

Subtract $30$ from $180$ .

$150$

Step 6.3

List the new angles.

$x = 150$

Step 7

The period of the $\tan (x)$ function is $180$ so values will repeat every $180$ degrees in both directions.

$x = 150 + 180 n, 150 + 180 n$ , for any integer $n$

Step 8

Consolidate the answers.

$x = 150 + 180 n$ , for any integer $n$