Precalculus Examples

$\sin (x) = - \frac{2 π}{3}$

Step 1

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

$x = \arcsin (- \frac{2 π}{3})$

Step 2

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.

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

Step 3

Add $2 π$ and $π$ .

$x = 3 π - \arcsin (- \frac{2 π}{3})$

Step 4

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

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Step 4.1

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

$\frac{2 π}{| b |}$

Step 4.2

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

$\frac{2 π}{| 1 |}$

Step 4.3

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

$\frac{2 π}{1}$

Step 4.4

Divide $2 π$ by $1$ .

$2 π$

Step 5

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

$x = \arcsin (- \frac{2 π}{3}) + 2 π n, 3 π - \arcsin (- \frac{2 π}{3}) + 2 π n$ , for any integer $n$