Trigonometry Examples

$\tan^{2} (x) - 4 = 0$

Step 1

Add $4$ to both sides of the equation.

$\tan^{2} (x) = 4$

Step 2

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

$\tan (x) = \pm \sqrt{4}$

Step 3

Simplify $\pm \sqrt{4}$ .

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Step 3.1

Rewrite $4$ as $2^{2}$ .

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

Step 3.2

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

$\tan (x) = \pm 2$

Step 4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.1

First, use the positive value of the $\pm$ to find the first solution.

$\tan (x) = 2$

Step 4.2

Next, use the negative value of the $\pm$ to find the second solution.

$\tan (x) = - 2$

Step 4.3

The complete solution is the result of both the positive and negative portions of the solution.

$\tan (x) = 2, - 2$

Step 5

Set up each of the solutions to solve for $x$ .

$\tan (x) = 2$

$\tan (x) = - 2$

Step 6

Solve for $x$ in $\tan (x) = 2$ .

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Step 6.1

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

$x = \arctan (2)$

Step 6.2

Simplify the right side.

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Step 6.2.1

Evaluate $\arctan (2)$ .

$x = 1.10714871$

Step 6.3

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.

$x = (3.14159265) + 1.10714871$

Step 6.4

Solve for $x$ .

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Step 6.4.1

Remove parentheses.

$x = 3.14159265 + 1.10714871$

Step 6.4.2

Remove parentheses.

$x = (3.14159265) + 1.10714871$

Step 6.4.3

Add $3.14159265$ and $1.10714871$ .

$x = 4.24874137$

Step 6.5

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

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Step 6.5.1

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

$\frac{π}{| b |}$

Step 6.5.2

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

$\frac{π}{| 1 |}$

Step 6.5.3

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

$\frac{π}{1}$

Step 6.5.4

Divide $π$ by $1$ .

$π$

Step 6.6

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

$x = 1.10714871 + π n, 4.24874137 + π n$ , for any integer $n$

Step 7

Solve for $x$ in $\tan (x) = - 2$ .

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Step 7.1

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

$x = \arctan (- 2)$

Step 7.2

Simplify the right side.

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Step 7.2.1

Evaluate $\arctan (- 2)$ .

$x = - 1.10714871$

Step 7.3

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

$x = - 1.10714871 - (3.14159265)$

Step 7.4

Simplify the expression to find the second solution.

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Step 7.4.1

Add $2 π$ to $- 1.10714871 - (3.14159265)$ .

$x = - 1.10714871 - (3.14159265) + 2 π$

Step 7.4.2

The resulting angle of $2.03444393$ is positive and coterminal with $- 1.10714871 - (3.14159265)$ .

$x = 2.03444393$

Step 7.5

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

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Step 7.5.1

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

$\frac{π}{| b |}$

Step 7.5.2

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

$\frac{π}{| 1 |}$

Step 7.5.3

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

$\frac{π}{1}$

Step 7.5.4

Divide $π$ by $1$ .

$π$

Step 7.6

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

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Step 7.6.1

Add $π$ to $- 1.10714871$ to find the positive angle.

$- 1.10714871 + π$

Step 7.6.2

Replace with decimal approximation.

$3.14159265 - 1.10714871$

Step 7.6.3

Subtract $1.10714871$ from $3.14159265$ .

$2.03444393$

Step 7.6.4

List the new angles.

$x = 2.03444393$

Step 7.7

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

$x = 2.03444393 + π n, 2.03444393 + π n$ , for any integer $n$

Step 8

List all of the solutions.

$x = 1.10714871 + π n, 4.24874137 + π n, 2.03444393 + π n, 2.03444393 + π n$ , for any integer $n$

Step 9

Consolidate the solutions.

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Step 9.1

Consolidate $1.10714871 + π n$ and $4.24874137 + π n$ to $1.10714871 + π n$ .

$x = 1.10714871 + π n, 2.03444393 + π n, 2.03444393 + π n$ , for any integer $n$

Step 9.2

Consolidate $2.03444393 + π n$ and $2.03444393 + π n$ to $2.03444393 + π n$ .

$x = 1.10714871 + π n, 2.03444393 + π n$ , for any integer $n$