Precalculus Examples

$\sin (x) = - \frac{\sqrt{3}}{2}$

Step 1

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

$\sin (x) \cdot \frac{2}{2} = - \frac{\sqrt{3}}{2}$

Step 2

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

$\sin (x) \cdot 2$

Step 3

Move $2$ to the left of $\sin (x)$ .

$\frac{2 \sin (x)}{2} = - \frac{\sqrt{3}}{2}$

Step 4

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

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Step 4.1

Divide $2$ by $2$ .

$\sin (x) \cdot 1 = - \frac{\sqrt{3}}{2}$

Step 4.2

Multiply $\sin (x)$ by $1$ .

$\sin (x) = - \frac{\sqrt{3}}{2}$

Step 5

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

$x = \arcsin (- \frac{\sqrt{3}}{2})$

Step 6

Simplify the right side.

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Step 6.1

The exact value of $\arcsin (- \frac{\sqrt{3}}{2})$ is $- \frac{π}{3}$ .

$x = - \frac{π}{3}$

Step 7

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.

$x = 2 π + \frac{π}{3} + π$

Step 8

Simplify the expression to find the second solution.

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Step 8.1

Subtract $2 π$ from $2 π + \frac{π}{3} + π$ .

$x = 2 π + \frac{π}{3} + π - 2 π$

Step 8.2

The resulting angle of $\frac{4 π}{3}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{3} + π$ .

$x = \frac{4 π}{3}$

Step 9

Find the period of $\sin (x)$ .

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Step 9.1

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

$\frac{2 π}{| b |}$

Step 9.2

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

$\frac{2 π}{| 1 |}$

Step 9.3

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

$\frac{2 π}{1}$

Step 9.4

Divide $2 π$ by $1$ .

$2 π$

Step 10

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

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Step 10.1

Add $2 π$ to $- \frac{π}{3}$ to find the positive angle.

$- \frac{π}{3} + 2 π$

Step 10.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$2 π \cdot \frac{3}{3} - \frac{π}{3}$

Step 10.3

Combine fractions.

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Step 10.3.1

Combine $2 π$ and $\frac{3}{3}$ .

$\frac{2 π \cdot 3}{3} - \frac{π}{3}$

Step 10.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 3 - π}{3}$

Step 10.4

Simplify the numerator.

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Step 10.4.1

Multiply $3$ by $2$ .

$\frac{6 π - π}{3}$

Step 10.4.2

Subtract $π$ from $6 π$ .

$\frac{5 π}{3}$

Step 10.5

List the new angles.

$x = \frac{5 π}{3}$

Step 11

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

$x = \frac{4 π}{3} + 2 π n, \frac{5 π}{3} + 2 π n$ , for any integer $n$