Trigonometry Examples

$\tan (\frac{x}{2} + \frac{π}{4}) = - 1$

Step 1

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

$\frac{x}{2} + \frac{π}{4} = \arctan (- 1)$

Step 2

Simplify the right side.

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Step 2.1

The exact value of $\arctan (- 1)$ is $- \frac{π}{4}$ .

$\frac{x}{2} + \frac{π}{4} = - \frac{π}{4}$

Step 3

Move all terms not containing $x$ to the right side of the equation.

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Step 3.1

Subtract $\frac{π}{4}$ from both sides of the equation.

$\frac{x}{2} = - \frac{π}{4} - \frac{π}{4}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{x}{2} = \frac{- π - π}{4}$

Step 3.3

Subtract $π$ from $- π$ .

$\frac{x}{2} = \frac{- 2 π}{4}$

Step 3.4

Cancel the common factor of $- 2$ and $4$ .

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Step 3.4.1

Factor $2$ out of $- 2 π$ .

$\frac{x}{2} = \frac{2 (- π)}{4}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $2$ out of $4$ .

$\frac{x}{2} = \frac{2 (- π)}{2 (2)}$

Step 3.4.2.2

Cancel the common factor.

$\frac{x}{2} = \frac{2 (- π)}{2 \cdot 2}$

Step 3.4.2.3

Rewrite the expression.

$\frac{x}{2} = \frac{- π}{2}$

Step 3.5

Move the negative in front of the fraction.

$\frac{x}{2} = - \frac{π}{2}$

Step 4

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$x = - π$

Step 5

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.

$\frac{x}{2} + \frac{π}{4} = - \frac{π}{4} - π$

Step 6

Simplify the expression to find the second solution.

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Step 6.1

Add $2 π$ to $- \frac{π}{4} - π$ .

$\frac{x}{2} + \frac{π}{4} = - \frac{π}{4} - π + 2 π$

Step 6.2

The resulting angle of $\frac{3 π}{4}$ is positive and coterminal with $- \frac{π}{4} - π$ .

$\frac{x}{2} + \frac{π}{4} = \frac{3 π}{4}$

Step 6.3

Solve for $x$ .

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Step 6.3.1

Move all terms not containing $x$ to the right side of the equation.

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Step 6.3.1.1

Subtract $\frac{π}{4}$ from both sides of the equation.

$\frac{x}{2} = \frac{3 π}{4} - \frac{π}{4}$

Step 6.3.1.2

Combine the numerators over the common denominator.

$\frac{x}{2} = \frac{3 π - π}{4}$

Step 6.3.1.3

Subtract $π$ from $3 π$ .

$\frac{x}{2} = \frac{2 π}{4}$

Step 6.3.1.4

Cancel the common factor of $2$ and $4$ .

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Step 6.3.1.4.1

Factor $2$ out of $2 π$ .

$\frac{x}{2} = \frac{2 (π)}{4}$

Step 6.3.1.4.2

Cancel the common factors.

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Step 6.3.1.4.2.1

Factor $2$ out of $4$ .

$\frac{x}{2} = \frac{2 π}{2 \cdot 2}$

Step 6.3.1.4.2.2

Cancel the common factor.

$\frac{x}{2} = \frac{2 π}{2 \cdot 2}$

Step 6.3.1.4.2.3

Rewrite the expression.

$\frac{x}{2} = \frac{π}{2}$

Step 6.3.2

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$x = π$

Step 7

Find the period of $\tan (\frac{x}{2} + \frac{π}{4})$ .

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Step 7.1

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

$\frac{π}{| b |}$

Step 7.2

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{π}{| \frac{1}{2} |}$

Step 7.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{π}{\frac{1}{2}}$

Step 7.4

Multiply the numerator by the reciprocal of the denominator.

$π \cdot 2$

Step 7.5

Move $2$ to the left of $π$ .

$2 π$

Step 8

Add $2 π$ to every negative angle to get positive angles.

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Step 8.1

Add $2 π$ to $- π$ to find the positive angle.

$- π + 2 π$

Step 8.2

Subtract $π$ from $2 π$ .

$π$

Step 8.3

List the new angles.

$x = π$

Step 9

The period of the $\tan (\frac{x}{2} + \frac{π}{4})$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

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

Step 10

Consolidate the answers.

$x = π + 2 π n$ , for any integer $n$