Trigonometry Examples

$9 \tan^{2} (x - 27) = 0$

Step 1

Divide each term in $9 \tan^{2} (x - 27) = 0$ by $9$ and simplify.

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Step 1.1

Divide each term in $9 \tan^{2} (x - 27) = 0$ by $9$ .

$\frac{9 \tan^{2} (x - 27)}{9} = \frac{0}{9}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $9$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{9 \tan^{2} (x - 27)}{9} = \frac{0}{9}$

Step 1.2.1.2

Divide $\tan^{2} (x - 27)$ by $1$ .

$\tan^{2} (x - 27) = \frac{0}{9}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $0$ by $9$ .

$\tan^{2} (x - 27) = 0$

Step 2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\tan (x - 27) = \pm \sqrt{0}$

Step 3

Simplify $\pm \sqrt{0}$ .

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Step 3.1

Rewrite $0$ as $0^{2}$ .

$\tan (x - 27) = \pm \sqrt{0^{2}}$

Step 3.2

Pull terms out from under the radical, assuming positive real numbers.

$\tan (x - 27) = \pm 0$

Step 3.3

Plus or minus $0$ is $0$ .

$\tan (x - 27) = 0$

Step 4

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

$x - 27 = \arctan (0)$

Step 5

Simplify the right side.

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Step 5.1

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

$x - 27 = 0$

Step 6

Add $27$ to both sides of the equation.

$x = 27$

Step 7

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.

$x - 27 = π + 0$

Step 8

Solve for $x$ .

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Step 8.1

Add $π$ and $0$ .

$x - 27 = π$

Step 8.2

Add $27$ to both sides of the equation.

$x = π + 27$

Step 9

Find the period of $\tan (x - 27)$ .

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Step 9.1

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

$\frac{π}{| b |}$

Step 9.2

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

$\frac{π}{| 1 |}$

Step 9.3

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

$\frac{π}{1}$

Step 9.4

Divide $π$ by $1$ .

$π$

Step 10

The period of the $\tan (x - 27)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = 27 + π n, π + 27 + π n$ , for any integer $n$