Trigonometry Examples

$\tan (x + 4) = 0$

Step 1

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

$x + 4 = \arctan (0)$

Step 2

Simplify the right side.

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

$x + 4 = 0$

Step 3

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 4

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$x + 4 = π + 0$

Step 5

Solve for $x$ .

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Add $π$ and $0$ .

$x + 4 = π$

Subtract $4$ from both sides of the equation.

$x = π - 4$

Step 6

Find the period of $\tan (x + 4)$ .

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

$\frac{π}{| b |}$

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

$\frac{π}{| 1 |}$

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

$\frac{π}{1}$

Divide $π$ by $1$ .

$π$

Step 7

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

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Add $π$ to $- 4$ to find the positive angle.

$- 4 + π$

List the new angles.

$x = - 4 + π$

Step 8

The period of the $\tan (x + 4)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = - 4 + π + π n, π - 4 + π + π n$ , for any integer $n$