Trigonometry Examples

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

Step 1

Subtract $3 \sin (x)$ from both sides of the equation.

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

Step 2

Simplify the left side of the equation.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite $\tan (x)$ in terms of sines and cosines.

${(\frac{\sin (x)}{\cos (x)})}^{2} \sin (x) - 3 \sin (x) = 0$

Step 2.1.2

Apply the product rule to $\frac{\sin (x)}{\cos (x)}$ .

$\frac{\sin^{2} (x)}{\cos^{2} (x)} \cdot \sin (x) - 3 \sin (x) = 0$

Step 2.1.3

Multiply $\frac{\sin^{2} (x)}{\cos^{2} (x)} \sin (x)$ .

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Step 2.1.3.1

Combine $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ and $\sin (x)$ .

$\frac{\sin^{2} (x) \sin (x)}{\cos^{2} (x)} - 3 \sin (x) = 0$

Step 2.1.3.2

Multiply $\sin^{2} (x)$ by $\sin (x)$ by adding the exponents.

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Step 2.1.3.2.1

Multiply $\sin^{2} (x)$ by $\sin (x)$ .

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Step 2.1.3.2.1.1

Raise $\sin (x)$ to the power of $1$ .

$\frac{\sin^{2} (x) \sin (x)}{\cos^{2} (x)} - 3 \sin (x) = 0$

Step 2.1.3.2.1.2

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

$\frac{{\sin (x)}^{2 + 1}}{\cos^{2} (x)} - 3 \sin (x) = 0$

Step 2.1.3.2.2

Add $2$ and $1$ .

$\frac{\sin^{3} (x)}{\cos^{2} (x)} - 3 \sin (x) = 0$

Step 2.2

Simplify each term.

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Step 2.2.1

Factor $\sin^{2} (x)$ out of $\sin^{3} (x)$ .

$\frac{\sin^{2} (x) \sin (x)}{\cos^{2} (x)} - 3 \sin (x) = 0$

Step 2.2.2

Multiply by $1$ .

$\frac{\sin^{2} (x) \sin (x)}{\cos^{2} (x) \cdot 1} - 3 \sin (x) = 0$

Step 2.2.3

Separate fractions.

$\frac{\sin (x)}{1} \cdot \frac{\sin^{2} (x)}{\cos^{2} (x)} - 3 \sin (x) = 0$

Step 2.2.4

Convert from $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ to $\tan^{2} (x)$ .

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

Step 2.2.5

Divide $\sin (x)$ by $1$ .

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

Step 3

Factor $\sin (x)$ out of $\sin (x) \tan^{2} (x) - 3 \sin (x)$ .

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Step 3.1

Factor $\sin (x)$ out of $- 3 \sin (x)$ .

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

Step 3.2

Factor $\sin (x)$ out of $\sin (x) \tan^{2} (x) + \sin (x) \cdot - 3$ .

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

Step 4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\sin (x) = 0$

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

Step 5

Set $\sin (x)$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $\sin (x)$ equal to $0$ .

$\sin (x) = 0$

Step 5.2

Solve $\sin (x) = 0$ for $x$ .

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Step 5.2.1

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

$x = \arcsin (0)$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

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

$x = 0$

Step 5.2.3

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$x = π - 0$

Step 5.2.4

Subtract $0$ from $π$ .

$x = π$

Step 5.2.5

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

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Step 5.2.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 5.2.5.2

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

$\frac{2 π}{| 1 |}$

Step 5.2.5.3

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

$\frac{2 π}{1}$

Step 5.2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.2.6

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

$x = 2 π n, π + 2 π n$ , for any integer $n$

Step 6

Set $\tan^{2} (x) - 3$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $\tan^{2} (x) - 3$ equal to $0$ .

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

Step 6.2

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

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Step 6.2.1

Add $3$ to both sides of the equation.

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

Step 6.2.2

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

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

Step 6.2.3

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

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Step 6.2.3.1

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

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

Step 6.2.3.2

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

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

Step 6.2.3.3

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

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

Step 6.2.4

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

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

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

Step 6.2.5

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

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Step 6.2.5.1

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

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

Step 6.2.5.2

Simplify the right side.

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Step 6.2.5.2.1

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

$x = \frac{π}{3}$

Step 6.2.5.3

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.

$x = π + \frac{π}{3}$

Step 6.2.5.4

Simplify $π + \frac{π}{3}$ .

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Step 6.2.5.4.1

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

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

Step 6.2.5.4.2

Combine fractions.

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Step 6.2.5.4.2.1

Combine $π$ and $\frac{3}{3}$ .

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

Step 6.2.5.4.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 3 + π}{3}$

Step 6.2.5.4.3

Simplify the numerator.

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Step 6.2.5.4.3.1

Move $3$ to the left of $π$ .

$x = \frac{3 \cdot π + π}{3}$

Step 6.2.5.4.3.2

Add $3 π$ and $π$ .

$x = \frac{4 π}{3}$

Step 6.2.5.5

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

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Step 6.2.5.5.1

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

$\frac{π}{| b |}$

Step 6.2.5.5.2

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

$\frac{π}{| 1 |}$

Step 6.2.5.5.3

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

$\frac{π}{1}$

Step 6.2.5.5.4

Divide $π$ by $1$ .

$π$

Step 6.2.5.6

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

$x = \frac{π}{3} + π n, \frac{4 π}{3} + π n$ , for any integer $n$

Step 6.2.6

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

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Step 6.2.6.1

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

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

Step 6.2.6.2

Simplify the right side.

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Step 6.2.6.2.1

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

$x = - \frac{π}{3}$

Step 6.2.6.3

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.

$x = - \frac{π}{3} - π$

Step 6.2.6.4

Simplify the expression to find the second solution.

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Step 6.2.6.4.1

Add $2 π$ to $- \frac{π}{3} - π$ .

$x = - \frac{π}{3} - π + 2 π$

Step 6.2.6.4.2

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

$x = \frac{2 π}{3}$

Step 6.2.6.5

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

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Step 6.2.6.5.1

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

$\frac{π}{| b |}$

Step 6.2.6.5.2

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

$\frac{π}{| 1 |}$

Step 6.2.6.5.3

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

$\frac{π}{1}$

Step 6.2.6.5.4

Divide $π$ by $1$ .

$π$

Step 6.2.6.6

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

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Step 6.2.6.6.1

Add $π$ to $- \frac{π}{3}$ to find the positive angle.

$- \frac{π}{3} + π$

Step 6.2.6.6.2

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

$π \cdot \frac{3}{3} - \frac{π}{3}$

Step 6.2.6.6.3

Combine fractions.

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Step 6.2.6.6.3.1

Combine $π$ and $\frac{3}{3}$ .

$\frac{π \cdot 3}{3} - \frac{π}{3}$

Step 6.2.6.6.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 3 - π}{3}$

Step 6.2.6.6.4

Simplify the numerator.

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Step 6.2.6.6.4.1

Move $3$ to the left of $π$ .

$\frac{3 \cdot π - π}{3}$

Step 6.2.6.6.4.2

Subtract $π$ from $3 π$ .

$\frac{2 π}{3}$

Step 6.2.6.6.5

List the new angles.

$x = \frac{2 π}{3}$

Step 6.2.6.7

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

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

Step 6.2.7

List all of the solutions.

$x = \frac{π}{3} + π n, \frac{4 π}{3} + π n, \frac{2 π}{3} + π n, \frac{2 π}{3} + π n$ , for any integer $n$

Step 6.2.8

Consolidate the solutions.

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Step 6.2.8.1

Consolidate $\frac{π}{3} + π n$ and $\frac{4 π}{3} + π n$ to $\frac{π}{3} + π n$ .

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

Step 6.2.8.2

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

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

Step 7

The final solution is all the values that make $\sin (x) (\tan^{2} (x) - 3) = 0$ true.

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

Step 8

Consolidate the answers.

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Step 8.1

Consolidate $2 π n$ and $π + 2 π n$ to $π n$ .

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

Step 8.2

Consolidate the answers.

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