Trigonometry Examples

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

Step 1

Subtract $\frac{π}{4}$ from each section of the inequality because it does not contain the variable we are trying to solve for.

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

Step 2

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

$- \frac{π}{2} \cdot \frac{2}{2} - \frac{π}{4} < x < \frac{π}{2} - \frac{π}{4}$

Step 3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{π}{2}$ by $\frac{2}{2}$ .

$- \frac{π \cdot 2}{2 \cdot 2} - \frac{π}{4} < x < \frac{π}{2} - \frac{π}{4}$

Step 3.2

Multiply $2$ by $2$ .

$- \frac{π \cdot 2}{4} - \frac{π}{4} < x < \frac{π}{2} - \frac{π}{4}$

Step 4

Combine the numerators over the common denominator.

$\frac{- π \cdot 2 - π}{4} < x < \frac{π}{2} - \frac{π}{4}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $2$ by $- 1$ .

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

Step 5.2

Subtract $π$ from $- 2 π$ .

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

Step 6

Move the negative in front of the fraction.

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

Step 7

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

$- \frac{3 π}{4} < x < \frac{π}{2} \cdot \frac{2}{2} - \frac{π}{4}$

Step 8

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.1

Multiply $\frac{π}{2}$ by $\frac{2}{2}$ .

$- \frac{3 π}{4} < x < \frac{π \cdot 2}{2 \cdot 2} - \frac{π}{4}$

Step 8.2

Multiply $2$ by $2$ .

$- \frac{3 π}{4} < x < \frac{π \cdot 2}{4} - \frac{π}{4}$

Step 9

Combine the numerators over the common denominator.

$- \frac{3 π}{4} < x < \frac{π \cdot 2 - π}{4}$

Step 10

Simplify the numerator.

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Step 10.1

Move $2$ to the left of $π$ .

$- \frac{3 π}{4} < x < \frac{2 \cdot π - π}{4}$

Step 10.2

Subtract $π$ from $2 π$ .

$- \frac{3 π}{4} < x < \frac{π}{4}$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$- \frac{3 π}{4} < x < \frac{π}{4}$

Interval Notation:

$(- \frac{3 π}{4}, \frac{π}{4})$

Step 12