# Trigonometry Examples

Let . Substitute for all occurrences of .

Factor by grouping.

For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .

Factor out of .

Rewrite as plus

Apply the distributive property.

Multiply by .

Remove parentheses.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

Factor out the greatest common factor (GCF) from each group.

Factor the polynomial by factoring out the greatest common factor, .

Replace all occurrences of with .

Replace the left side with the factored expression.

If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .

Set the first factor equal to .

Add to both sides of the equation.

Simplify the expression to find the first solution.

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

The exact value of is .

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

Simplify the expression to find the second solution.

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 the expression .

Multiply by .

Add and .

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.

Consolidate the answers.

Set the next factor equal to .

Subtract from both sides of the equation.

Divide each term by and simplify.

Divide each term in by .

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Divide by .

Simplify the expression to find the first solution.

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

Evaluate .

The tangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from to find the solution in the third quadrant.

Simplify the expression to find the second solution.

Multiply by .

Subtract from .

Add to .

The resulting angle of is positive 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.

Subtract from .

List the new angles.

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

The final solution is all the values that make true.