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 $π$ .

$\frac{x}{2} = \frac{0}{4}$

Step 3.4

Divide $0$ by $4$ .

$\frac{x}{2} = 0$

Step 4

Set the numerator equal to zero.

$x = 0$

Step 5

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.

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

Step 6

Solve for $x$ .

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Step 6.1

Simplify $π + \frac{π}{4}$ .

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Step 6.1.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 6.1.2

Combine fractions.

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Step 6.1.2.1

Combine $π$ and $\frac{4}{4}$ .

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

Step 6.1.2.2

Combine the numerators over the common denominator.

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

Step 6.1.3

Simplify the numerator.

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Step 6.1.3.1

Move $4$ to the left of $π$ .

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

Step 6.1.3.2

Add $4 π$ and $π$ .

$\frac{x}{2} + \frac{π}{4} = \frac{5 π}{4}$

Step 6.2

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

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Step 6.2.1

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

$\frac{x}{2} = \frac{5 π}{4} - \frac{π}{4}$

Step 6.2.2

Combine the numerators over the common denominator.

$\frac{x}{2} = \frac{5 π - π}{4}$

Step 6.2.3

Subtract $π$ from $5 π$ .

$\frac{x}{2} = \frac{4 π}{4}$

Step 6.2.4

Cancel the common factor of $4$ .

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Step 6.2.4.1

Cancel the common factor.

$\frac{x}{2} = \frac{4 π}{4}$

Step 6.2.4.2

Divide $π$ by $1$ .

$\frac{x}{2} = π$

Step 6.3

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 π$

Step 6.4

Simplify the left side.

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Step 6.4.1

Cancel the common factor of $2$ .

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Step 6.4.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 π$

Step 6.4.1.2

Rewrite the expression.

$x = 2 π$

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

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.

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

Step 9

Consolidate the answers.

$x = 2 π n$ , for any integer $n$