Trigonometry Examples

$6 \csc (θ) + 7 = 0$

Step 1

Subtract $7$ from both sides of the equation.

$6 \csc (θ) = - 7$

Step 2

Divide each term in $6 \csc (θ) = - 7$ by $6$ and simplify.

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Step 2.1

Divide each term in $6 \csc (θ) = - 7$ by $6$ .

$\frac{6 \csc (θ)}{6} = \frac{- 7}{6}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $6$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{6 \csc (θ)}{6} = \frac{- 7}{6}$

Step 2.2.1.2

Divide $\csc (θ)$ by $1$ .

$\csc (θ) = \frac{- 7}{6}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

$\csc (θ) = - \frac{7}{6}$

Step 3

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

$θ = arccsc (- \frac{7}{6})$

Step 4

Simplify the right side.

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Step 4.1

Evaluate $arccsc (- \frac{7}{6})$ .

$θ = - 58.99728086$

Step 5

The cosecant function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $360$ , to find a reference angle. Next, add this reference angle to $180$ to find the solution in the third quadrant.

$θ = 360 + 58.99728086 + 180$

Step 6

Simplify the expression to find the second solution.

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Step 6.1

Subtract $360 °$ from $360 + 58.99728086 + 180 °$ .

$θ = 360 + 58.99728086 + 180 ° - 360 °$

Step 6.2

The resulting angle of $238.99728086 °$ is positive, less than $360 °$ , and coterminal with $360 + 58.99728086 + 180$ .

$θ = 238.99728086 °$

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

Add $360$ to every negative angle to get positive angles.

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Step 8.1

Add $360$ to $- 58.99728086$ to find the positive angle.

$- 58.99728086 + 360$

Step 8.2

Subtract $58.99728086$ from $360$ .

$301.00271913$

Step 8.3

List the new angles.

$θ = 301.00271913$

Step 9

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

$θ = 238.99728086 + 360 n, 301.00271913 + 360 n$ , for any integer $n$