Trigonometry Examples

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

Step 1

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

$2 (u) = 1 - {(u)}^{2}$

Step 2

Add $u^{2}$ to both sides of the equation.

$2 u + u^{2} = 1$

Step 3

Subtract $1$ from both sides of the equation.

$2 u + u^{2} - 1 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 1$ , $b = 2$ , and $c = - 1$ into the quadratic formula and solve for $u$ .

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

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

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

Step 6.1.2

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

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

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

Step 6.1.2.2

Multiply $- 4$ by $- 1$ .

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

Step 6.1.3

Add $4$ and $4$ .

$u = \frac{- 2 \pm \sqrt{8}}{2 \cdot 1}$

Step 6.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 6.1.4.1

Factor $4$ out of $8$ .

$u = \frac{- 2 \pm \sqrt{4 (2)}}{2 \cdot 1}$

Step 6.1.4.2

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

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

Step 6.1.5

Pull terms out from under the radical.

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

Step 6.2

Multiply $2$ by $1$ .

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

Step 6.3

Simplify $\frac{- 2 \pm 2 \sqrt{2}}{2}$ .

$u = - 1 \pm \sqrt{2}$

Step 7

The final answer is the combination of both solutions.

$u = - 1 + \sqrt{2}, - 1 - \sqrt{2}$

Step 8

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

$\tan (x) = - 1 + \sqrt{2}, - 1 - \sqrt{2}$

Step 9

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

$\tan (x) = - 1 + \sqrt{2}$

$\tan (x) = - 1 - \sqrt{2}$

Step 10

Solve for $x$ in $\tan (x) = - 1 + \sqrt{2}$ .

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Step 10.1

Convert the right side of the equation to its decimal equivalent.

$\tan (x) = 0.41421356$

Step 10.2

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

$x = \arctan (0.41421356)$

Step 10.3

Simplify the right side.

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Step 10.3.1

Evaluate $\arctan (0.41421356)$ .

$x = \frac{π}{8}$

Step 10.4

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.

$x = π + \frac{π}{8}$

Step 10.5

Simplify $π + \frac{π}{8}$ .

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Step 10.5.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$x = π \cdot \frac{8}{8} + \frac{π}{8}$

Step 10.5.2

Combine fractions.

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Step 10.5.2.1

Combine $π$ and $\frac{8}{8}$ .

$x = \frac{π \cdot 8}{8} + \frac{π}{8}$

Step 10.5.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 8 + π}{8}$

Step 10.5.3

Simplify the numerator.

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Step 10.5.3.1

Move $8$ to the left of $π$ .

$x = \frac{8 \cdot π + π}{8}$

Step 10.5.3.2

Add $8 π$ and $π$ .

$x = \frac{9 π}{8}$

Step 10.6

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

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Step 10.6.1

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

$\frac{π}{| b |}$

Step 10.6.2

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

$\frac{π}{| 1 |}$

Step 10.6.3

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

$\frac{π}{1}$

Step 10.6.4

Divide $π$ by $1$ .

$π$

Step 10.7

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

$x = \frac{π}{8} + π n, \frac{9 π}{8} + π n$ , for any integer $n$

Step 11

Solve for $x$ in $\tan (x) = - 1 - \sqrt{2}$ .

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Step 11.1

Convert the right side of the equation to its decimal equivalent.

$\tan (x) = - 2.41421356$

Step 11.2

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

$x = \arctan (- 2.41421356)$

Step 11.3

Simplify the right side.

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Step 11.3.1

Evaluate $\arctan (- 2.41421356)$ .

$x = - \frac{3 π}{8}$

Step 11.4

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.

$x = - \frac{3 π}{8} - π$

Step 11.5

Simplify the expression to find the second solution.

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Step 11.5.1

Add $2 π$ to $- \frac{3 π}{8} - π$ .

$x = - \frac{3 π}{8} - π + 2 π$

Step 11.5.2

The resulting angle of $\frac{5 π}{8}$ is positive and coterminal with $- \frac{3 π}{8} - π$ .

$x = \frac{5 π}{8}$

Step 11.6

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

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Step 11.6.1

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

$\frac{π}{| b |}$

Step 11.6.2

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

$\frac{π}{| 1 |}$

Step 11.6.3

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

$\frac{π}{1}$

Step 11.6.4

Divide $π$ by $1$ .

$π$

Step 11.7

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

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Step 11.7.1

Add $π$ to $- \frac{3 π}{8}$ to find the positive angle.

$- \frac{3 π}{8} + π$

Step 11.7.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$π \cdot \frac{8}{8} - \frac{3 π}{8}$

Step 11.7.3

Combine fractions.

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Step 11.7.3.1

Combine $π$ and $\frac{8}{8}$ .

$\frac{π \cdot 8}{8} - \frac{3 π}{8}$

Step 11.7.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 8 - 3 π}{8}$

Step 11.7.4

Simplify the numerator.

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Step 11.7.4.1

Move $8$ to the left of $π$ .

$\frac{8 \cdot π - 3 π}{8}$

Step 11.7.4.2

Subtract $3 π$ from $8 π$ .

$\frac{5 π}{8}$

Step 11.7.5

List the new angles.

$x = \frac{5 π}{8}$

Step 11.8

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

$x = \frac{5 π}{8} + π n, \frac{5 π}{8} + π n$ , for any integer $n$

Step 12

List all of the solutions.

$x = \frac{π}{8} + π n, \frac{9 π}{8} + π n, \frac{5 π}{8} + π n, \frac{5 π}{8} + π n$ , for any integer $n$

Step 13

Consolidate the answers.

$x = \frac{π}{8} + \frac{π n}{2}$ , for any integer $n$