Trigonometry Examples

$y = 1 - 2 \sec (\frac{1}{2} x + π)$

Step 1

Set the argument in $\sec (\frac{1}{2} x + π)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$\frac{1}{2} x + π = \frac{π}{2} + π n$ , for any integer $n$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

Combine $\frac{1}{2}$ and $x$ .

$\frac{x}{2} + π = \frac{π}{2} + π n$

Step 2.2

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

Tap for more steps...

Step 2.2.1

Subtract $π$ from both sides of the equation.

$\frac{x}{2} = \frac{π}{2} + π n - π$

Step 2.2.2

To write $- π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x}{2} = π n + \frac{π}{2} - π \cdot \frac{2}{2}$

Step 2.2.3

Combine $- π$ and $\frac{2}{2}$ .

$\frac{x}{2} = π n + \frac{π}{2} + \frac{- π \cdot 2}{2}$

Step 2.2.4

Combine the numerators over the common denominator.

$\frac{x}{2} = π n + \frac{π - π \cdot 2}{2}$

Step 2.2.5

Simplify each term.

Tap for more steps...

Step 2.2.5.1

Simplify the numerator.

Tap for more steps...

Step 2.2.5.1.1

Multiply $2$ by $- 1$ .

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

Step 2.2.5.1.2

Subtract $2 π$ from $π$ .

$\frac{x}{2} = π n + \frac{- π}{2}$

Step 2.2.5.2

Move the negative in front of the fraction.

$\frac{x}{2} = π n - \frac{π}{2}$

Step 2.3

Multiply both sides of the equation by $2$ .

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

Step 2.4

Simplify both sides of the equation.

Tap for more steps...

Step 2.4.1

Simplify the left side.

Tap for more steps...

Step 2.4.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.4.1.1.1

Cancel the common factor.

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

Step 2.4.1.1.2

Rewrite the expression.

$x = 2 (π n - \frac{π}{2})$

Step 2.4.2

Simplify the right side.

Tap for more steps...

Step 2.4.2.1

Simplify $2 (π n - \frac{π}{2})$ .

Tap for more steps...

Step 2.4.2.1.1

Apply the distributive property.

$x = 2 (π n) + 2 (- \frac{π}{2})$

Step 2.4.2.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.4.2.1.2.1

Move the leading negative in $- \frac{π}{2}$ into the numerator.

$x = 2 π n + 2 \frac{- π}{2}$

Step 2.4.2.1.2.2

Cancel the common factor.

$x = 2 π n + 2 \frac{- π}{2}$

Step 2.4.2.1.2.3

Rewrite the expression.

$x = 2 π n - π$

Step 3

The domain is all values of $x$ that make the expression defined.

Set-Builder Notation:

${x | x \neq 2 π n - π}$ , for any integer $n$

Step 4