Precalculus Examples

$\sin (θ) = \frac{\sqrt{3}}{2}$

Step 1

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

$θ = \arcsin (\frac{\sqrt{3}}{2})$

Step 2

Simplify the right side.

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Step 2.1

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

$θ = \frac{π}{3}$

Step 3

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$θ = π - \frac{π}{3}$

Step 4

Simplify $π - \frac{π}{3}$ .

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Step 4.1

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

$θ = π \cdot \frac{3}{3} - \frac{π}{3}$

Step 4.2

Combine fractions.

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Step 4.2.1

Combine $π$ and $\frac{3}{3}$ .

$θ = \frac{π \cdot 3}{3} - \frac{π}{3}$

Step 4.2.2

Combine the numerators over the common denominator.

$θ = \frac{π \cdot 3 - π}{3}$

Step 4.3

Simplify the numerator.

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Step 4.3.1

Move $3$ to the left of $π$ .

$θ = \frac{3 \cdot π - π}{3}$

Step 4.3.2

Subtract $π$ from $3 π$ .

$θ = \frac{2 π}{3}$

Step 5

Find the period of $\sin (θ)$ .

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Step 5.1

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

$\frac{2 π}{| b |}$

Step 5.2

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

$\frac{2 π}{| 1 |}$

Step 5.3

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

$\frac{2 π}{1}$

Step 5.4

Divide $2 π$ by $1$ .

$2 π$

Step 6

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

$θ = \frac{π}{3} + 2 π n, \frac{2 π}{3} + 2 π n$ , for any integer $n$