Trigonometry Examples

$\csc (θ) = \frac{2 \sqrt{3}}{3}$

Step 1

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

$θ = arccsc (\frac{2 \sqrt{3}}{3})$

Step 2

Simplify the right side.

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Step 2.1

The exact value of $arccsc (\frac{2 \sqrt{3}}{3})$ is $60$ .

$θ = 60$

Step 3

The cosecant function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the second quadrant.

$θ = 180 - 60$

Step 4

Subtract $60$ from $180$ .

$θ = 120$

Step 5

Find the period of $\csc (θ)$ .

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Step 5.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 5.2

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

$\frac{360}{| 1 |}$

Step 5.3

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

$\frac{360}{1}$

Step 5.4

Divide $360$ by $1$ .

$360$

Step 6

The period of the $\csc (θ)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$θ = 60 + 360 n, 120 + 360 n$ , for any integer $n$