# Trigonometry Examples

Step 1

Step 1.1

Let . Substitute for all occurrences of .

Step 1.2

Factor by grouping.

Step 1.2.1

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

Step 1.2.1.1

Factor out of .

Step 1.2.1.2

Rewrite as plus

Step 1.2.1.3

Apply the distributive property.

Step 1.2.2

Factor out the greatest common factor from each group.

Step 1.2.2.1

Group the first two terms and the last two terms.

Step 1.2.2.2

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

Step 1.2.3

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

Step 1.3

Replace all occurrences of with .

Step 2

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

Step 3

Step 3.1

Set equal to .

Step 3.2

Solve for .

Step 3.2.1

Add to both sides of the equation.

Step 3.2.2

Divide each term in by and simplify.

Step 3.2.2.1

Divide each term in by .

Step 3.2.2.2

Simplify the left side.

Step 3.2.2.2.1

Cancel the common factor of .

Step 3.2.2.2.1.1

Cancel the common factor.

Step 3.2.2.2.1.2

Divide by .

Step 3.2.3

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

Step 3.2.4

Simplify the right side.

Step 3.2.4.1

Evaluate .

Step 3.2.5

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

Step 3.2.6

Solve for .

Step 3.2.6.1

Remove parentheses.

Step 3.2.6.2

Simplify .

Step 3.2.6.2.1

Multiply by .

Step 3.2.6.2.2

Subtract from .

Step 3.2.7

Find the period of .

Step 3.2.7.1

The period of the function can be calculated using .

Step 3.2.7.2

Replace with in the formula for period.

Step 3.2.7.3

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

Step 3.2.7.4

Divide by .

Step 3.2.8

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

, for any integer

, for any integer

, for any integer

Step 4

Step 4.1

Set equal to .

Step 4.2

Solve for .

Step 4.2.1

Subtract from both sides of the equation.

Step 4.2.2

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

Step 4.2.3

Simplify the right side.

Step 4.2.3.1

The exact value of is .

Step 4.2.4

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

Step 4.2.5

Subtract from .

Step 4.2.6

Find the period of .

Step 4.2.6.1

The period of the function can be calculated using .

Step 4.2.6.2

Replace with in the formula for period.

Step 4.2.6.3

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

Step 4.2.6.4

Divide by .

Step 4.2.7

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

, for any integer

, for any integer

, for any integer

Step 5

The final solution is all the values that make true.

, for any integer