Trigonometry Examples

$\tan^{2} (2 x) - 3 = 0$

Step 1

Add $3$ to both sides of the equation.

$\tan^{2} (2 x) = 3$

Step 2

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

$\tan (2 x) = \pm \sqrt{3}$

Step 3

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

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Step 3.1

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

$\tan (2 x) = \sqrt{3}$

Step 3.2

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

$\tan (2 x) = - \sqrt{3}$

Step 3.3

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

$\tan (2 x) = \sqrt{3}, - \sqrt{3}$

Step 4

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

$\tan (2 x) = \sqrt{3}$

$\tan (2 x) = - \sqrt{3}$

Step 5

Solve for $x$ in $\tan (2 x) = \sqrt{3}$ .

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Step 5.1

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

$2 x = \arctan (\sqrt{3})$

Step 5.2

Simplify the right side.

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Step 5.2.1

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

$2 x = \frac{π}{3}$

Step 5.3

Divide each term in $2 x = \frac{π}{3}$ by $2$ and simplify.

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Step 5.3.1

Divide each term in $2 x = \frac{π}{3}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{π}{3}}{2}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $2$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{π}{3}}{2}$

Step 5.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{3}}{2}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{3} \cdot \frac{1}{2}$

Step 5.3.3.2

Multiply $\frac{π}{3} \cdot \frac{1}{2}$ .

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Step 5.3.3.2.1

Multiply $\frac{π}{3}$ by $\frac{1}{2}$ .

$x = \frac{π}{3 \cdot 2}$

Step 5.3.3.2.2

Multiply $3$ by $2$ .

$x = \frac{π}{6}$

Step 5.4

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.

$2 x = π + \frac{π}{3}$

Step 5.5

Solve for $x$ .

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Step 5.5.1

Simplify.

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Step 5.5.1.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$2 x = π \cdot \frac{3}{3} + \frac{π}{3}$

Step 5.5.1.2

Combine $π$ and $\frac{3}{3}$ .

$2 x = \frac{π \cdot 3}{3} + \frac{π}{3}$

Step 5.5.1.3

Combine the numerators over the common denominator.

$2 x = \frac{π \cdot 3 + π}{3}$

Step 5.5.1.4

Add $π \cdot 3$ and $π$ .

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Step 5.5.1.4.1

Reorder $π$ and $3$ .

$2 x = \frac{3 \cdot π + π}{3}$

Step 5.5.1.4.2

Add $3 \cdot π$ and $π$ .

$2 x = \frac{4 \cdot π}{3}$

Step 5.5.2

Divide each term in $2 x = \frac{4 \cdot π}{3}$ by $2$ and simplify.

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Step 5.5.2.1

Divide each term in $2 x = \frac{4 \cdot π}{3}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{4 \cdot π}{3}}{2}$

Step 5.5.2.2

Simplify the left side.

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Step 5.5.2.2.1

Cancel the common factor of $2$ .

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Step 5.5.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{4 \cdot π}{3}}{2}$

Step 5.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{4 \cdot π}{3}}{2}$

Step 5.5.2.3

Simplify the right side.

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Step 5.5.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{4 \cdot π}{3} \cdot \frac{1}{2}$

Step 5.5.2.3.2

Cancel the common factor of $2$ .

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Step 5.5.2.3.2.1

Factor $2$ out of $4 \cdot π$ .

$x = \frac{2 (2 \cdot π)}{3} \cdot \frac{1}{2}$

Step 5.5.2.3.2.2

Cancel the common factor.

$x = \frac{2 (2 \cdot π)}{3} \cdot \frac{1}{2}$

Step 5.5.2.3.2.3

Rewrite the expression.

$x = \frac{2 \cdot π}{3}$

$x = \frac{2 π}{3}$

Step 5.6

Find the period of $\tan (2 x)$ .

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Step 5.6.1

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

$\frac{π}{| b |}$

Step 5.6.2

Replace $b$ with $2$ in the formula for period.

$\frac{π}{| 2 |}$

Step 5.6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{π}{2}$

Step 5.7

The period of the $\tan (2 x)$ function is $\frac{π}{2}$ so values will repeat every $\frac{π}{2}$ radians in both directions.

$x = \frac{π}{6} + \frac{π n}{2}, \frac{2 π}{3} + \frac{π n}{2}$ , for any integer $n$

Step 6

Solve for $x$ in $\tan (2 x) = - \sqrt{3}$ .

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Step 6.1

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

$2 x = \arctan (- \sqrt{3})$

Step 6.2

Simplify the right side.

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Step 6.2.1

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

$2 x = - \frac{π}{3}$

Step 6.3

Divide each term in $2 x = - \frac{π}{3}$ by $2$ and simplify.

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Step 6.3.1

Divide each term in $2 x = - \frac{π}{3}$ by $2$ .

$\frac{2 x}{2} = \frac{- \frac{π}{3}}{2}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $2$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- \frac{π}{3}}{2}$

Step 6.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{π}{3}}{2}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{π}{3} \cdot \frac{1}{2}$

Step 6.3.3.2

Multiply $- \frac{π}{3} \cdot \frac{1}{2}$ .

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Step 6.3.3.2.1

Multiply $\frac{1}{2}$ by $\frac{π}{3}$ .

$x = - \frac{π}{2 \cdot 3}$

Step 6.3.3.2.2

Multiply $2$ by $3$ .

$x = - \frac{π}{6}$

Step 6.4

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.

$2 x = - \frac{π}{3} - π$

Step 6.5

Simplify the expression to find the second solution.

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Step 6.5.1

Add $2 π$ to $- \frac{π}{3} - π$ .

$2 x = - \frac{π}{3} - π + 2 π$

Step 6.5.2

The resulting angle of $\frac{2 π}{3}$ is positive and coterminal with $- \frac{π}{3} - π$ .

$2 x = \frac{2 π}{3}$

Step 6.5.3

Divide each term in $2 x = \frac{2 π}{3}$ by $2$ and simplify.

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Step 6.5.3.1

Divide each term in $2 x = \frac{2 π}{3}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{2 π}{3}}{2}$

Step 6.5.3.2

Simplify the left side.

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Step 6.5.3.2.1

Cancel the common factor of $2$ .

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Step 6.5.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{2 π}{3}}{2}$

Step 6.5.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{2 π}{3}}{2}$

Step 6.5.3.3

Simplify the right side.

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Step 6.5.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{2 π}{3} \cdot \frac{1}{2}$

Step 6.5.3.3.2

Cancel the common factor of $2$ .

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Step 6.5.3.3.2.1

Factor $2$ out of $2 π$ .

$x = \frac{2 (π)}{3} \cdot \frac{1}{2}$

Step 6.5.3.3.2.2

Cancel the common factor.

$x = \frac{2 π}{3} \cdot \frac{1}{2}$

Step 6.5.3.3.2.3

Rewrite the expression.

$x = \frac{π}{3}$

Step 6.6

Find the period of $\tan (2 x)$ .

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Step 6.6.1

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

$\frac{π}{| b |}$

Step 6.6.2

Replace $b$ with $2$ in the formula for period.

$\frac{π}{| 2 |}$

Step 6.6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{π}{2}$

Step 6.7

Add $\frac{π}{2}$ to every negative angle to get positive angles.

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Step 6.7.1

Add $\frac{π}{2}$ to $- \frac{π}{6}$ to find the positive angle.

$- \frac{π}{6} + \frac{π}{2}$

Step 6.7.2

To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{π}{2} \cdot \frac{3}{3} - \frac{π}{6}$

Step 6.7.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.7.3.1

Multiply $\frac{π}{2}$ by $\frac{3}{3}$ .

$\frac{π \cdot 3}{2 \cdot 3} - \frac{π}{6}$

Step 6.7.3.2

Multiply $2$ by $3$ .

$\frac{π \cdot 3}{6} - \frac{π}{6}$

Step 6.7.4

Combine the numerators over the common denominator.

$\frac{π \cdot 3 - π}{6}$

Step 6.7.5

Simplify the numerator.

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Step 6.7.5.1

Move $3$ to the left of $π$ .

$\frac{3 \cdot π - π}{6}$

Step 6.7.5.2

Subtract $π$ from $3 π$ .

$\frac{2 π}{6}$

Step 6.7.6

Cancel the common factor of $2$ and $6$ .

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Step 6.7.6.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{6}$

Step 6.7.6.2

Cancel the common factors.

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Step 6.7.6.2.1

Factor $2$ out of $6$ .

$\frac{2 π}{2 \cdot 3}$

Step 6.7.6.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 3}$

Step 6.7.6.2.3

Rewrite the expression.

$\frac{π}{3}$

Step 6.7.7

List the new angles.

$x = \frac{π}{3}$

Step 6.8

The period of the $\tan (2 x)$ function is $\frac{π}{2}$ so values will repeat every $\frac{π}{2}$ radians in both directions.

$x = \frac{π}{3} + \frac{π n}{2}, \frac{π}{3} + \frac{π n}{2}$ , for any integer $n$

Step 7

List all of the solutions.

$x = \frac{π}{6} + \frac{π n}{2}, \frac{2 π}{3} + \frac{π n}{2}, \frac{π}{3} + \frac{π n}{2}, \frac{π}{3} + \frac{π n}{2}$ , for any integer $n$

Step 8

Consolidate the solutions.

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Step 8.1

Consolidate $\frac{π}{6} + \frac{π n}{2}$ and $\frac{2 π}{3} + \frac{π n}{2}$ to $\frac{π}{6} + \frac{π n}{2}$ .

$x = \frac{π}{6} + \frac{π n}{2}, \frac{π}{3} + \frac{π n}{2}, \frac{π}{3} + \frac{π n}{2}$ , for any integer $n$

Step 8.2

Consolidate $\frac{π}{3} + \frac{π n}{2}$ and $\frac{π}{3} + \frac{π n}{2}$ to $\frac{π}{3} + \frac{π n}{2}$ .

$x = \frac{π}{6} + \frac{π n}{2}, \frac{π}{3} + \frac{π n}{2}$ , for any integer $n$