Trigonometry Examples

$\sin (x) = \frac{5 π}{3}$

Step 1

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

$x = \arcsin (\frac{5 π}{3})$

Step 2

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.

$x = π - \arcsin (\frac{5 π}{3})$

Step 3

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

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Step 3.1

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

$\frac{2 π}{| b |}$

Step 3.2

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

$\frac{2 π}{| 1 |}$

Step 3.3

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

$\frac{2 π}{1}$

Step 3.4

Divide $2 π$ by $1$ .

$2 π$

Step 4

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

$x = \arcsin (\frac{5 π}{3}) + 2 π n, π - \arcsin (\frac{5 π}{3}) + 2 π n$ , for any integer $n$