Trigonometry Examples

$\tan^{4} (x) - 20 \tan^{2} (x) + 64 = 0$

Step 1

Factor $\tan^{4} (x) - 20 \tan^{2} (x) + 64$ .

Tap for more steps...

Step 1.1

Rewrite $\tan^{4} (x)$ as ${(\tan^{2} (x))}^{2}$ .

${(\tan^{2} (x))}^{2} - 20 \tan^{2} (x) + 64 = 0$

Step 1.2

Let $u = \tan^{2} (x)$ . Substitute $u$ for all occurrences of $\tan^{2} (x)$ .

$u^{2} - 20 u + 64 = 0$

Step 1.3

Factor $u^{2} - 20 u + 64$ using the AC method.

Tap for more steps...

Step 1.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $64$ and whose sum is $- 20$ .

$- 16, - 4$

Step 1.3.2

Write the factored form using these integers.

$(u - 16) (u - 4) = 0$

Step 1.4

Replace all occurrences of $u$ with $\tan^{2} (x)$ .

$(\tan^{2} (x) - 16) (\tan^{2} (x) - 4) = 0$

Step 1.5

Rewrite $16$ as $4^{2}$ .

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

Step 1.6

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \tan (x)$ and $b = 4$ .

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

Step 1.7

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

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

Step 1.8

Factor.

Tap for more steps...

Step 1.8.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \tan (x)$ and $b = 2$ .

$(\tan (x) + 4) (\tan (x) - 4) ((\tan (x) + 2) (\tan (x) - 2)) = 0$

Step 1.8.2

Remove unnecessary parentheses.

$(\tan (x) + 4) (\tan (x) - 4) (\tan (x) + 2) (\tan (x) - 2) = 0$

Step 2

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

$\tan (x) + 4 = 0$

$\tan (x) - 4 = 0$

$\tan (x) + 2 = 0$

$\tan (x) - 2 = 0$

Step 3

Set $\tan (x) + 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.1

Set $\tan (x) + 4$ equal to $0$ .

$\tan (x) + 4 = 0$

Step 3.2

Solve $\tan (x) + 4 = 0$ for $x$ .

Tap for more steps...

Step 3.2.1

Subtract $4$ from both sides of the equation.

$\tan (x) = - 4$

Step 3.2.2

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

$x = \arctan (- 4)$

Step 3.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.3.1

Evaluate $\arctan (- 4)$ .

$x = - 1.32581766$

Step 3.2.4

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.

$x = - 1.32581766 - (3.14159265)$

Step 3.2.5

Simplify the expression to find the second solution.

Tap for more steps...

Step 3.2.5.1

Add $2 π$ to $- 1.32581766 - (3.14159265)$ .

$x = - 1.32581766 - (3.14159265) + 2 π$

Step 3.2.5.2

The resulting angle of $1.81577498$ is positive and coterminal with $- 1.32581766 - (3.14159265)$ .

$x = 1.81577498$

Step 3.2.6

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

Tap for more steps...

Step 3.2.6.1

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

$\frac{π}{| b |}$

Step 3.2.6.2

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

$\frac{π}{| 1 |}$

Step 3.2.6.3

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

$\frac{π}{1}$

Step 3.2.6.4

Divide $π$ by $1$ .

$π$

Step 3.2.7

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

Tap for more steps...

Step 3.2.7.1

Add $π$ to $- 1.32581766$ to find the positive angle.

$- 1.32581766 + π$

Step 3.2.7.2

Replace with decimal approximation.

$3.14159265 - 1.32581766$

Step 3.2.7.3

Subtract $1.32581766$ from $3.14159265$ .

$1.81577498$

Step 3.2.7.4

List the new angles.

$x = 1.81577498$

Step 3.2.8

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

$x = 1.81577498 + π n, 1.81577498 + π n$ , for any integer $n$

Step 4

Set $\tan (x) - 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.1

Set $\tan (x) - 4$ equal to $0$ .

$\tan (x) - 4 = 0$

Step 4.2

Solve $\tan (x) - 4 = 0$ for $x$ .

Tap for more steps...

Step 4.2.1

Add $4$ to both sides of the equation.

$\tan (x) = 4$

Step 4.2.2

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

$x = \arctan (4)$

Step 4.2.3

Simplify the right side.

Tap for more steps...

Step 4.2.3.1

Evaluate $\arctan (4)$ .

$x = 1.32581766$

Step 4.2.4

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.

$x = (3.14159265) + 1.32581766$

Step 4.2.5

Solve for $x$ .

Tap for more steps...

Step 4.2.5.1

Remove parentheses.

$x = 3.14159265 + 1.32581766$

Step 4.2.5.2

Remove parentheses.

$x = (3.14159265) + 1.32581766$

Step 4.2.5.3

Add $3.14159265$ and $1.32581766$ .

$x = 4.46741031$

Step 4.2.6

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

Tap for more steps...

Step 4.2.6.1

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

$\frac{π}{| b |}$

Step 4.2.6.2

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

$\frac{π}{| 1 |}$

Step 4.2.6.3

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

$\frac{π}{1}$

Step 4.2.6.4

Divide $π$ by $1$ .

$π$

Step 4.2.7

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

$x = 1.32581766 + π n, 4.46741031 + π n$ , for any integer $n$

Step 5

Set $\tan (x) + 2$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 5.1

Set $\tan (x) + 2$ equal to $0$ .

$\tan (x) + 2 = 0$

Step 5.2

Solve $\tan (x) + 2 = 0$ for $x$ .

Tap for more steps...

Step 5.2.1

Subtract $2$ from both sides of the equation.

$\tan (x) = - 2$

Step 5.2.2

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

$x = \arctan (- 2)$

Step 5.2.3

Simplify the right side.

Tap for more steps...

Step 5.2.3.1

Evaluate $\arctan (- 2)$ .

$x = - 1.10714871$

Step 5.2.4

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.

$x = - 1.10714871 - (3.14159265)$

Step 5.2.5

Simplify the expression to find the second solution.

Tap for more steps...

Step 5.2.5.1

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

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

Step 5.2.5.2

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

$x = 2.03444393$

Step 5.2.6

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

Tap for more steps...

Step 5.2.6.1

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

$\frac{π}{| b |}$

Step 5.2.6.2

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

$\frac{π}{| 1 |}$

Step 5.2.6.3

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

$\frac{π}{1}$

Step 5.2.6.4

Divide $π$ by $1$ .

$π$

Step 5.2.7

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

Tap for more steps...

Step 5.2.7.1

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

$- 1.10714871 + π$

Step 5.2.7.2

Replace with decimal approximation.

$3.14159265 - 1.10714871$

Step 5.2.7.3

Subtract $1.10714871$ from $3.14159265$ .

$2.03444393$

Step 5.2.7.4

List the new angles.

$x = 2.03444393$

Step 5.2.8

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 6

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

Tap for more steps...

Step 6.1

Set $\tan (x) - 2$ equal to $0$ .

$\tan (x) - 2 = 0$

Step 6.2

Solve $\tan (x) - 2 = 0$ for $x$ .

Tap for more steps...

Step 6.2.1

Add $2$ to both sides of the equation.

$\tan (x) = 2$

Step 6.2.2

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

$x = \arctan (2)$

Step 6.2.3

Simplify the right side.

Tap for more steps...

Step 6.2.3.1

Evaluate $\arctan (2)$ .

$x = 1.10714871$

Step 6.2.4

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.

$x = (3.14159265) + 1.10714871$

Step 6.2.5

Solve for $x$ .

Tap for more steps...

Step 6.2.5.1

Remove parentheses.

$x = 3.14159265 + 1.10714871$

Step 6.2.5.2

Remove parentheses.

$x = (3.14159265) + 1.10714871$

Step 6.2.5.3

Add $3.14159265$ and $1.10714871$ .

$x = 4.24874137$

Step 6.2.6

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

Tap for more steps...

Step 6.2.6.1

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

$\frac{π}{| b |}$

Step 6.2.6.2

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

$\frac{π}{| 1 |}$

Step 6.2.6.3

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

$\frac{π}{1}$

Step 6.2.6.4

Divide $π$ by $1$ .

$π$

Step 6.2.7

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

The final solution is all the values that make $(\tan (x) + 4) (\tan (x) - 4) (\tan (x) + 2) (\tan (x) - 2) = 0$ true.

$x = 1.81577498 + π n, 1.81577498 + π n, 1.32581766 + π n, 4.46741031 + π n, 2.03444393 + π n, 2.03444393 + π n, 1.10714871 + π n, 4.24874137 + π n$ , for any integer $n$

Step 8

Consolidate the answers.

Tap for more steps...

Step 8.1

Consolidate $1.81577498 + π n$ and $1.81577498 + π n$ to $1.81577498 + π n$ .

$x = 1.81577498 + π n, 1.32581766 + π n, 4.46741031 + π n, 2.03444393 + π n, 2.03444393 + π n, 1.10714871 + π n, 4.24874137 + π n$ , for any integer $n$

Step 8.2

Consolidate $1.32581766 + π n$ and $4.46741031 + π n$ to $1.32581766 + π n$ .

$x = 1.81577498 + π n, 1.32581766 + π n, 2.03444393 + π n, 2.03444393 + π n, 1.10714871 + π n, 4.24874137 + π n$ , for any integer $n$

Step 8.3

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

$x = 1.81577498 + π n, 1.32581766 + π n, 2.03444393 + π n, 1.10714871 + π n, 4.24874137 + π n$ , for any integer $n$

Step 8.4

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

$x = 1.81577498 + π n, 1.32581766 + π n, 2.03444393 + π n, 1.10714871 + π n$ , for any integer $n$