Trigonometry Examples

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

Step 1

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

${(u)}^{2} - 3 u + 1 = 0$

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 $a = 1$ , $b = - 3$ , and $c = 1$ into the quadratic formula and solve for $u$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 3$ to the power of $2$ .

$u = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 4.1.2

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

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{3 \pm \sqrt{9 - 4 \cdot 1}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $1$ .

$u = \frac{3 \pm \sqrt{9 - 4}}{2 \cdot 1}$

Step 4.1.3

Subtract $4$ from $9$ .

$u = \frac{3 \pm \sqrt{5}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$u = \frac{3 \pm \sqrt{5}}{2}$

Step 5

The final answer is the combination of both solutions.

$u = \frac{3 + \sqrt{5}}{2}, \frac{3 - \sqrt{5}}{2}$

Step 6

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

$\tan (x) = \frac{3 + \sqrt{5}}{2}, \frac{3 - \sqrt{5}}{2}$

Step 7

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

$\tan (x) = \frac{3 + \sqrt{5}}{2}$

$\tan (x) = \frac{3 - \sqrt{5}}{2}$

Step 8

Solve for $x$ in $\tan (x) = \frac{3 + \sqrt{5}}{2}$ .

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Step 8.1

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

$x = \arctan (\frac{3 + \sqrt{5}}{2})$

Step 8.2

Simplify the right side.

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Step 8.2.1

Evaluate $\arctan (\frac{3 + \sqrt{5}}{2})$ .

$x = 1.20593249$

Step 8.3

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.

$x = (3.14159265) + 1.20593249$

Step 8.4

Solve for $x$ .

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Step 8.4.1

Remove parentheses.

$x = 3.14159265 + 1.20593249$

Step 8.4.2

Remove parentheses.

$x = (3.14159265) + 1.20593249$

Step 8.4.3

Add $3.14159265$ and $1.20593249$ .

$x = 4.34752515$

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

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

$x = 1.20593249 + π n, 4.34752515 + π n$ , for any integer $n$

Step 9

Solve for $x$ in $\tan (x) = \frac{3 - \sqrt{5}}{2}$ .

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Step 9.1

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

$x = \arctan (\frac{3 - \sqrt{5}}{2})$

Step 9.2

Simplify the right side.

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Step 9.2.1

Evaluate $\arctan (\frac{3 - \sqrt{5}}{2})$ .

$x = 0.36486382$

Step 9.3

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.

$x = (3.14159265) + 0.36486382$

Step 9.4

Solve for $x$ .

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Step 9.4.1

Remove parentheses.

$x = 3.14159265 + 0.36486382$

Step 9.4.2

Remove parentheses.

$x = (3.14159265) + 0.36486382$

Step 9.4.3

Add $3.14159265$ and $0.36486382$ .

$x = 3.50645648$

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

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

$x = 0.36486382 + π n, 3.50645648 + π n$ , for any integer $n$

Step 10

List all of the solutions.

$x = 1.20593249 + π n, 4.34752515 + π n, 0.36486382 + π n, 3.50645648 + π n$ , for any integer $n$

Step 11

Consolidate the solutions.

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Step 11.1

Consolidate $1.20593249 + π n$ and $4.34752515 + π n$ to $1.20593249 + π n$ .

$x = 1.20593249 + π n, 0.36486382 + π n, 3.50645648 + π n$ , for any integer $n$

Step 11.2

Consolidate $0.36486382 + π n$ and $3.50645648 + π n$ to $0.36486382 + π n$ .

$x = 1.20593249 + π n, 0.36486382 + π n$ , for any integer $n$