Trigonometry Examples

$3 \tan^{2} (2 x) - 1 = 0$

Step 1

Add $1$ to both sides of the equation.

$3 \tan^{2} (2 x) = 1$

Step 2

Divide each term in $3 \tan^{2} (2 x) = 1$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 \tan^{2} (2 x) = 1$ by $3$ .

$\frac{3 \tan^{2} (2 x)}{3} = \frac{1}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3 \tan^{2} (2 x)}{3} = \frac{1}{3}$

Step 2.2.1.2

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

$\tan^{2} (2 x) = \frac{1}{3}$

Step 3

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

$\tan (2 x) = \pm \sqrt{\frac{1}{3}}$

Step 4

Simplify $\pm \sqrt{\frac{1}{3}}$ .

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Step 4.1

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$\tan (2 x) = \pm \frac{\sqrt{1}}{\sqrt{3}}$

Step 4.2

Any root of $1$ is $1$ .

$\tan (2 x) = \pm \frac{1}{\sqrt{3}}$

Step 4.3

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\tan (2 x) = \pm \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 4.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$\tan (2 x) = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 4.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$\tan (2 x) = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 4.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\tan (2 x) = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 4.4.5

Add $1$ and $1$ .

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

Step 4.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\tan (2 x) = \pm \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 4.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\tan (2 x) = \pm \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 4.4.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 4.4.6.4

Cancel the common factor of $2$ .

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Step 4.4.6.4.1

Cancel the common factor.

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

Step 4.4.6.4.2

Rewrite the expression.

$\tan (2 x) = \pm \frac{\sqrt{3}}{3^{1}}$

Step 4.4.6.5

Evaluate the exponent.

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

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

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Simplify the right side.

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Step 7.2.1

The exact value of $\arctan (\frac{\sqrt{3}}{3})$ is $\frac{π}{6}$ .

$2 x = \frac{π}{6}$

Step 7.3

Divide each term in $2 x = \frac{π}{6}$ by $2$ and simplify.

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Step 7.3.1

Divide each term in $2 x = \frac{π}{6}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{π}{6}}{2}$

Step 7.3.2

Simplify the left side.

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Step 7.3.2.1

Cancel the common factor of $2$ .

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Step 7.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{π}{6}}{2}$

Step 7.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{6}}{2}$

Step 7.3.3

Simplify the right side.

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Step 7.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{6} \cdot \frac{1}{2}$

Step 7.3.3.2

Multiply $\frac{π}{6} \cdot \frac{1}{2}$ .

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Step 7.3.3.2.1

Multiply $\frac{π}{6}$ by $\frac{1}{2}$ .

$x = \frac{π}{6 \cdot 2}$

Step 7.3.3.2.2

Multiply $6$ by $2$ .

$x = \frac{π}{12}$

Step 7.4

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.

$2 x = π + \frac{π}{6}$

Step 7.5

Solve for $x$ .

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Step 7.5.1

Simplify.

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Step 7.5.1.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$2 x = π \cdot \frac{6}{6} + \frac{π}{6}$

Step 7.5.1.2

Combine $π$ and $\frac{6}{6}$ .

$2 x = \frac{π \cdot 6}{6} + \frac{π}{6}$

Step 7.5.1.3

Combine the numerators over the common denominator.

$2 x = \frac{π \cdot 6 + π}{6}$

Step 7.5.1.4

Add $π \cdot 6$ and $π$ .

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Step 7.5.1.4.1

Reorder $π$ and $6$ .

$2 x = \frac{6 \cdot π + π}{6}$

Step 7.5.1.4.2

Add $6 \cdot π$ and $π$ .

$2 x = \frac{7 \cdot π}{6}$

Step 7.5.2

Divide each term in $2 x = \frac{7 \cdot π}{6}$ by $2$ and simplify.

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Step 7.5.2.1

Divide each term in $2 x = \frac{7 \cdot π}{6}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{7 \cdot π}{6}}{2}$

Step 7.5.2.2

Simplify the left side.

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Step 7.5.2.2.1

Cancel the common factor of $2$ .

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Step 7.5.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{7 \cdot π}{6}}{2}$

Step 7.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{7 \cdot π}{6}}{2}$

Step 7.5.2.3

Simplify the right side.

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Step 7.5.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{7 \cdot π}{6} \cdot \frac{1}{2}$

Step 7.5.2.3.2

Multiply $\frac{7 π}{6} \cdot \frac{1}{2}$ .

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Step 7.5.2.3.2.1

Multiply $\frac{7 π}{6}$ by $\frac{1}{2}$ .

$x = \frac{7 π}{6 \cdot 2}$

Step 7.5.2.3.2.2

Multiply $6$ by $2$ .

$x = \frac{7 π}{12}$

Step 7.6

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

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Step 7.6.1

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

$\frac{π}{| b |}$

Step 7.6.2

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

$\frac{π}{| 2 |}$

Step 7.6.3

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

$\frac{π}{2}$

Step 7.7

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

$x = \frac{π}{12} + \frac{π n}{2}, \frac{7 π}{12} + \frac{π n}{2}$ , for any integer $n$

Step 8

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

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Step 8.1

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

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

Step 8.2

Simplify the right side.

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Step 8.2.1

The exact value of $\arctan (- \frac{\sqrt{3}}{3})$ is $- \frac{π}{6}$ .

$2 x = - \frac{π}{6}$

Step 8.3

Divide each term in $2 x = - \frac{π}{6}$ by $2$ and simplify.

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Step 8.3.1

Divide each term in $2 x = - \frac{π}{6}$ by $2$ .

$\frac{2 x}{2} = \frac{- \frac{π}{6}}{2}$

Step 8.3.2

Simplify the left side.

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Step 8.3.2.1

Cancel the common factor of $2$ .

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Step 8.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- \frac{π}{6}}{2}$

Step 8.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{π}{6}}{2}$

Step 8.3.3

Simplify the right side.

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Step 8.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{π}{6} \cdot \frac{1}{2}$

Step 8.3.3.2

Multiply $- \frac{π}{6} \cdot \frac{1}{2}$ .

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Step 8.3.3.2.1

Multiply $\frac{1}{2}$ by $\frac{π}{6}$ .

$x = - \frac{π}{2 \cdot 6}$

Step 8.3.3.2.2

Multiply $2$ by $6$ .

$x = - \frac{π}{12}$

Step 8.4

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.

$2 x = - \frac{π}{6} - π$

Step 8.5

Simplify the expression to find the second solution.

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Step 8.5.1

Add $2 π$ to $- \frac{π}{6} - π$ .

$2 x = - \frac{π}{6} - π + 2 π$

Step 8.5.2

The resulting angle of $\frac{5 π}{6}$ is positive and coterminal with $- \frac{π}{6} - π$ .

$2 x = \frac{5 π}{6}$

Step 8.5.3

Divide each term in $2 x = \frac{5 π}{6}$ by $2$ and simplify.

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Step 8.5.3.1

Divide each term in $2 x = \frac{5 π}{6}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{5 π}{6}}{2}$

Step 8.5.3.2

Simplify the left side.

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Step 8.5.3.2.1

Cancel the common factor of $2$ .

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Step 8.5.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{5 π}{6}}{2}$

Step 8.5.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{5 π}{6}}{2}$

Step 8.5.3.3

Simplify the right side.

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Step 8.5.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{5 π}{6} \cdot \frac{1}{2}$

Step 8.5.3.3.2

Multiply $\frac{5 π}{6} \cdot \frac{1}{2}$ .

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Step 8.5.3.3.2.1

Multiply $\frac{5 π}{6}$ by $\frac{1}{2}$ .

$x = \frac{5 π}{6 \cdot 2}$

Step 8.5.3.3.2.2

Multiply $6$ by $2$ .

$x = \frac{5 π}{12}$

Step 8.6

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

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Step 8.6.1

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

$\frac{π}{| b |}$

Step 8.6.2

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

$\frac{π}{| 2 |}$

Step 8.6.3

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

$\frac{π}{2}$

Step 8.7

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

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Step 8.7.1

Add $\frac{π}{2}$ to $- \frac{π}{12}$ to find the positive angle.

$- \frac{π}{12} + \frac{π}{2}$

Step 8.7.2

To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{π}{2} \cdot \frac{6}{6} - \frac{π}{12}$

Step 8.7.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.7.3.1

Multiply $\frac{π}{2}$ by $\frac{6}{6}$ .

$\frac{π \cdot 6}{2 \cdot 6} - \frac{π}{12}$

Step 8.7.3.2

Multiply $2$ by $6$ .

$\frac{π \cdot 6}{12} - \frac{π}{12}$

Step 8.7.4

Combine the numerators over the common denominator.

$\frac{π \cdot 6 - π}{12}$

Step 8.7.5

Simplify the numerator.

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Step 8.7.5.1

Move $6$ to the left of $π$ .

$\frac{6 \cdot π - π}{12}$

Step 8.7.5.2

Subtract $π$ from $6 π$ .

$\frac{5 π}{12}$

Step 8.7.6

List the new angles.

$x = \frac{5 π}{12}$

Step 8.8

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

$x = \frac{5 π}{12} + \frac{π n}{2}, \frac{5 π}{12} + \frac{π n}{2}$ , for any integer $n$

Step 9

List all of the solutions.

$x = \frac{π}{12} + \frac{π n}{2}, \frac{7 π}{12} + \frac{π n}{2}, \frac{5 π}{12} + \frac{π n}{2}, \frac{5 π}{12} + \frac{π n}{2}$ , for any integer $n$

Step 10

Consolidate the solutions.

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Step 10.1

Consolidate $\frac{π}{12} + \frac{π n}{2}$ and $\frac{7 π}{12} + \frac{π n}{2}$ to $\frac{π}{12} + \frac{π n}{2}$ .

$x = \frac{π}{12} + \frac{π n}{2}, \frac{5 π}{12} + \frac{π n}{2}, \frac{5 π}{12} + \frac{π n}{2}$ , for any integer $n$

Step 10.2

Consolidate $\frac{5 π}{12} + \frac{π n}{2}$ and $\frac{5 π}{12} + \frac{π n}{2}$ to $\frac{5 π}{12} + \frac{π n}{2}$ .

$x = \frac{π}{12} + \frac{π n}{2}, \frac{5 π}{12} + \frac{π n}{2}$ , for any integer $n$