Trigonometry Examples

$2 \csc (θ) - 3 = 0$

Step 1

Add $3$ to both sides of the equation.

$2 \csc (θ) = 3$

Step 2

Divide each term in $2 \csc (θ) = 3$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 \csc (θ) = 3$ by $2$ .

$\frac{2 \csc (θ)}{2} = \frac{3}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 \csc (θ)}{2} = \frac{3}{2}$

Step 2.2.1.2

Divide $\csc (θ)$ by $1$ .

$\csc (θ) = \frac{3}{2}$

Step 3

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

$θ = arccsc (\frac{3}{2})$

Step 4

Simplify the right side.

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Step 4.1

Evaluate $arccsc (\frac{3}{2})$ .

$θ = 41.81031489$

Step 5

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.

$θ = 180 - 41.81031489$

Step 6

Subtract $41.81031489$ from $180$ .

$θ = 138.1896851$

Step 7

Find the period of $\csc (θ)$ .

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

$\frac{360}{1}$

Step 7.4

Divide $360$ by $1$ .

$360$

Step 8

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

$θ = 41.81031489 + 360 n, 138.1896851 + 360 n$ , for any integer $n$