Subtract <math><mstyle displaystyle="true"><mn>6</mn></mstyle></math> from both sides of the equation.

Subtract <math><mstyle displaystyle="true"><mn>6</mn></mstyle></math> from <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn></mstyle></math> .

Divide each term in <math><mstyle displaystyle="true"><mn>4</mn><mi>csc</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow><mo>=</mo><mo>-</mo><mn>8</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Simplify the left side.

Cancel the common factor of <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Cancel the common factor.

Divide <math><mstyle displaystyle="true"><mi>csc</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Simplify the right side.

Divide <math><mstyle displaystyle="true"><mo>-</mo><mn>8</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Take the inverse cosecant of both sides of the equation to extract <math><mstyle displaystyle="true"><mi>θ</mi></mstyle></math> from inside the cosecant.

The exact value of <math><mstyle displaystyle="true"><mi>arccsc</mi><mrow><mo>(</mo><mo>-</mo><mn>2</mn><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mo>-</mo><mn>30</mn></mstyle></math> .

The cosecant function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from <math><mstyle displaystyle="true"><mn>360</mn></mstyle></math> , to find a reference angle. Next, add this reference angle to <math><mstyle displaystyle="true"><mn>180</mn></mstyle></math> to find the solution in the third quadrant.

Subtract <math><mstyle displaystyle="true"><mn>360</mn><mi>°</mi></mstyle></math> from <math><mstyle displaystyle="true"><mn>360</mn><mo>+</mo><mn>30</mn><mo>+</mo><mn>180</mn><mi>°</mi></mstyle></math> .

The resulting angle of <math><mstyle displaystyle="true"><mn>210</mn><mi>°</mi></mstyle></math> is positive, less than <math><mstyle displaystyle="true"><mn>360</mn><mi>°</mi></mstyle></math> , and coterminal with <math><mstyle displaystyle="true"><mn>360</mn><mo>+</mo><mn>30</mn><mo>+</mo><mn>180</mn></mstyle></math> .

The period of the function can be calculated using <math><mstyle displaystyle="true"><mfrac><mrow><mn>360</mn></mrow><mrow><mrow><mo>|</mo><mi>b</mi><mo>|</mo></mrow></mrow></mfrac></mstyle></math> .

Replace <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> with <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> in the formula for period.

The absolute value is the distance between a number and zero. The distance between <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> is <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Divide <math><mstyle displaystyle="true"><mn>360</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>360</mn></mstyle></math> to <math><mstyle displaystyle="true"><mo>-</mo><mn>30</mn></mstyle></math> to find the positive angle.

Subtract <math><mstyle displaystyle="true"><mn>30</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>360</mn></mstyle></math> .

List the new angles.

The period of the <math><mstyle displaystyle="true"><mi>csc</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></mstyle></math> function is <math><mstyle displaystyle="true"><mn>360</mn></mstyle></math> so values will repeat every <math><mstyle displaystyle="true"><mn>360</mn></mstyle></math> degrees in both directions.

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