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.

Tap for more steps...

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.

Tap for more steps...

Step 3.1

Add $\frac{π}{4}$ to 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

Add $π$ and $π$ .

$\frac{x}{2} = \frac{2 π}{4}$

Step 3.4

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

Tap for more steps...

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.

Tap for more steps...

Step 3.4.2.1

Factor $2$ out of $4$ .

$\frac{x}{2} = \frac{2 π}{2 \cdot 2}$

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 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 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 $x$ .

Tap for more steps...

Step 6.1

Simplify $π + \frac{π}{4}$ .

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

Step 6.2.1

Add $\frac{π}{4}$ to 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

Add $5 π$ and $π$ .

$\frac{x}{2} = \frac{6 π}{4}$

Step 6.2.4

Cancel the common factor of $6$ and $4$ .

Tap for more steps...

Step 6.2.4.1

Factor $2$ out of $6 π$ .

$\frac{x}{2} = \frac{2 (3 π)}{4}$

Step 6.2.4.2

Cancel the common factors.

Tap for more steps...

Step 6.2.4.2.1

Factor $2$ out of $4$ .

$\frac{x}{2} = \frac{2 (3 π)}{2 (2)}$

Step 6.2.4.2.2

Cancel the common factor.

$\frac{x}{2} = \frac{2 (3 π)}{2 \cdot 2}$

Step 6.2.4.2.3

Rewrite the expression.

$\frac{x}{2} = \frac{3 π}{2}$

Step 6.3

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

$x = 3 π$

Step 7

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

Tap for more steps...

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.

$x = π + 2 π n, 3 π + 2 π n$ , for any integer $n$

Step 9

Consolidate the answers.

$x = π + 2 π n$ , for any integer $n$