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# Trigonometry Examples

,

Add to both sides of the equation.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Simplify .

Multiply by .

Combine and simplify the denominator.

Multiply and .

Raise to the power of .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Rewrite as .

Rewrite as .

Apply the power rule and multiply exponents, .

Combine and .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Evaluate the exponent.

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

The exact value of is .

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

To write as a fraction with a common denominator, multiply by .

Write each expression with a common denominator of , by multiplying each by an appropriate factor of .

Combine.

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

Move to the left of .

Subtract from .

The period of the function can be calculated using .

Replace with in the formula for period.

Solve the equation.

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

Divide by .

The period of the function is so values will repeat every radians in both directions.

, for any integer

Plug in for and simplify to see if the solution is contained in .

Plug in for .

Simplify.

Multiply .

Multiply by .

Multiply by .

Add and .

The interval contains .

Plug in for and simplify to see if the solution is contained in .

Plug in for .

Simplify.

Multiply .

Multiply by .

Multiply by .

Add and .

The interval contains .