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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 .

Take the root of both sides of the to eliminate the exponent on the left side.

Simplify the right side of the equation.

Rewrite as .

Simplify the denominator.

Rewrite as .

Pull terms out from under the radical, assuming positive real numbers.

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the to find the first solution.

Next, use the negative value of the to find the second solution.

The complete solution is the result of both the positive and negative portions of the solution.

Set up each of the solutions to solve for .

Set up the equation to solve for .

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

The exact value of is .

The sine 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.

Simplify .

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 .

Find the period.

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

, for any integer

Set up the equation to solve for .

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

The exact value of is .

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

Simplify the expression to find the second solution.

Simplify .

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.

Multiply by .

Add and .

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 .

Add and .

Subtract from .

The resulting angle of is positive, less than , and coterminal with .

Find the period.

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 .

Add to every negative angle to get positive angles.

Add to to find the positive angle.

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

Combine.

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply by .

Subtract from .

List the new angles.

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

, for any integer

, for any integer

List all of the results found in the previous steps.

, for any integer

Consolidate and to .

, for any integer

Consolidate and to .

, for any integer

, for any integer

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

Plug in for .

Simplify.

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 by .

Add and .

The interval contains .

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

Plug in for .

Simplify.

Multiply by .

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

Combine.

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

Move to the left of .

Add and .

The interval contains .

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

Plug in for .

Simplify.

Multiply by .

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

Combine.

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

Move to the left of .

Add and .

The interval contains .