Trigonometry Examples

$\tan^{2} (x) + 9 \tan (x) + 1 = 0$

Step 1

Substitute $u$ for $\tan (x)$ .

${(u)}^{2} + 9 (u) + 1 = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 1$ , $b = 9$ , and $c = 1$ into the quadratic formula and solve for $u$ .

$\frac{- 9 \pm \sqrt{9^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $9$ to the power of $2$ .

$u = \frac{- 9 \pm \sqrt{81 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{- 9 \pm \sqrt{81 - 4 \cdot 1}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $1$ .

$u = \frac{- 9 \pm \sqrt{81 - 4}}{2 \cdot 1}$

Step 4.1.3

Subtract $4$ from $81$ .

$u = \frac{- 9 \pm \sqrt{77}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$u = \frac{- 9 \pm \sqrt{77}}{2}$

Step 5

The final answer is the combination of both solutions.

$u = - \frac{9 - \sqrt{77}}{2}, - \frac{9 + \sqrt{77}}{2}$

Step 6

Substitute $\tan (x)$ for $u$ .

$\tan (x) = - \frac{9 - \sqrt{77}}{2}, - \frac{9 + \sqrt{77}}{2}$

Step 7

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

$\tan (x) = - \frac{9 - \sqrt{77}}{2}$

$\tan (x) = - \frac{9 + \sqrt{77}}{2}$

Step 8

Solve for $x$ in $\tan (x) = - \frac{9 - \sqrt{77}}{2}$ .

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Step 8.1

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

$x = \arctan (- \frac{9 - \sqrt{77}}{2})$

Step 8.2

Simplify the right side.

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Step 8.2.1

Evaluate $\arctan (- \frac{9 - \sqrt{77}}{2})$ .

$x = - 0.11204654$

Step 8.3

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.

$x = - 0.11204654 - (3.14159265)$

Step 8.4

Simplify the expression to find the second solution.

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Step 8.4.1

Add $2 π$ to $- 0.11204654 - (3.14159265)$ .

$x = - 0.11204654 - (3.14159265) + 2 π$

Step 8.4.2

The resulting angle of $3.0295461$ is positive and coterminal with $- 0.11204654 - (3.14159265)$ .

$x = 3.0295461$

Step 8.5

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

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Step 8.5.1

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

$\frac{π}{| b |}$

Step 8.5.2

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

$\frac{π}{| 1 |}$

Step 8.5.3

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

$\frac{π}{1}$

Step 8.5.4

Divide $π$ by $1$ .

$π$

Step 8.6

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

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Step 8.6.1

Add $π$ to $- 0.11204654$ to find the positive angle.

$- 0.11204654 + π$

Step 8.6.2

Replace with decimal approximation.

$3.14159265 - 0.11204654$

Step 8.6.3

Subtract $0.11204654$ from $3.14159265$ .

$3.0295461$

Step 8.6.4

List the new angles.

$x = 3.0295461$

Step 8.7

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

$x = 3.0295461 + π n, 3.0295461 + π n$ , for any integer $n$

Step 9

Solve for $x$ in $\tan (x) = - \frac{9 + \sqrt{77}}{2}$ .

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Step 9.1

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

$x = \arctan (- \frac{9 + \sqrt{77}}{2})$

Step 9.2

Simplify the right side.

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Step 9.2.1

Evaluate $\arctan (- \frac{9 + \sqrt{77}}{2})$ .

$x = - 1.45874978$

Step 9.3

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.

$x = - 1.45874978 - (3.14159265)$

Step 9.4

Simplify the expression to find the second solution.

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Step 9.4.1

Add $2 π$ to $- 1.45874978 - (3.14159265)$ .

$x = - 1.45874978 - (3.14159265) + 2 π$

Step 9.4.2

The resulting angle of $1.68284287$ is positive and coterminal with $- 1.45874978 - (3.14159265)$ .

$x = 1.68284287$

Step 9.5

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

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Step 9.5.1

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

$\frac{π}{| b |}$

Step 9.5.2

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

$\frac{π}{| 1 |}$

Step 9.5.3

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

$\frac{π}{1}$

Step 9.5.4

Divide $π$ by $1$ .

$π$

Step 9.6

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

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Step 9.6.1

Add $π$ to $- 1.45874978$ to find the positive angle.

$- 1.45874978 + π$

Step 9.6.2

Replace with decimal approximation.

$3.14159265 - 1.45874978$

Step 9.6.3

Subtract $1.45874978$ from $3.14159265$ .

$1.68284287$

Step 9.6.4

List the new angles.

$x = 1.68284287$

Step 9.7

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

$x = 1.68284287 + π n, 1.68284287 + π n$ , for any integer $n$

Step 10

List all of the solutions.

$x = 3.0295461 + π n, 3.0295461 + π n, 1.68284287 + π n, 1.68284287 + π n$ , for any integer $n$

Step 11

Consolidate the solutions.

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Step 11.1

Consolidate $3.0295461 + π n$ and $3.0295461 + π n$ to $3.0295461 + π n$ .

$x = 3.0295461 + π n, 1.68284287 + π n, 1.68284287 + π n$ , for any integer $n$

Step 11.2

Consolidate $1.68284287 + π n$ and $1.68284287 + π n$ to $1.68284287 + π n$ .

$x = 3.0295461 + π n, 1.68284287 + π n$ , for any integer $n$