Precalculus Examples

$\sin (θ) = - \frac{\sqrt{2}}{2}$

Multiply each term by a factor of $1$ that will equate all the denominators. In this case, all terms need a denominator of $2$ .

$\sin (θ) \cdot \frac{2}{2} = - \frac{\sqrt{2}}{2}$

Multiply the expression by a factor of $1$ to create the least common denominator (LCD) of $2$ .

$\sin (θ) \cdot 2$

Move $2$ to the left of $\sin (θ)$ .

$\frac{2 \sin (θ)}{2} = - \frac{\sqrt{2}}{2}$

Simplify $\sin (θ) \cdot \frac{2}{2}$ .

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Divide $2$ by $2$ .

$\sin (θ) \cdot 1 = - \frac{\sqrt{2}}{2}$

Multiply $\sin (θ)$ by $1$ .

$\sin (θ) = - \frac{\sqrt{2}}{2}$

Take the inverse sine of both sides of the equation to extract $θ$ from inside the sine.

$θ = \arcsin (- \frac{\sqrt{2}}{2})$

Simplify the right side.

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The exact value of $\arcsin (- \frac{\sqrt{2}}{2})$ is $- \frac{π}{4}$ .

$θ = - \frac{π}{4}$

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$θ = 2 π + \frac{π}{4} + π$

Simplify the expression to find the second solution.

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Subtract $2 π$ from $2 π + \frac{π}{4} + π$ .

$θ = 2 π + \frac{π}{4} + π - 2 π$

The resulting angle of $\frac{5 π}{4}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{4} + π$ .

$θ = \frac{5 π}{4}$

Find the period of $\sin (θ)$ .

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The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

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

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Add $2 π$ to $- \frac{π}{4}$ to find the positive angle.

$- \frac{π}{4} + 2 π$

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$2 π \cdot \frac{4}{4} - \frac{π}{4}$

Combine fractions.

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Combine $2 π$ and $\frac{4}{4}$ .

$\frac{2 π \cdot 4}{4} - \frac{π}{4}$

Combine the numerators over the common denominator.

$\frac{2 π \cdot 4 - π}{4}$

Simplify the numerator.

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Multiply $4$ by $2$ .

$\frac{8 π - π}{4}$

Subtract $π$ from $8 π$ .

$\frac{7 π}{4}$

List the new angles.

$θ = \frac{7 π}{4}$

The period of the $\sin (θ)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$θ = \frac{5 π}{4} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer $n$