# Trigonometry Examples

Step 1

Add to both sides of the equation.

Step 2

Step 2.1

Divide each term in by .

Step 2.2

Simplify the left side.

Step 2.2.1

Cancel the common factor of .

Step 2.2.1.1

Cancel the common factor.

Step 2.2.1.2

Divide by .

Step 3

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

Step 4

Step 4.1

Rewrite as .

Step 4.2

Any root of is .

Step 4.3

Simplify the denominator.

Step 4.3.1

Rewrite as .

Step 4.3.2

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

Step 5

Step 5.1

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

Step 5.2

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

Step 5.3

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

Step 6

Set up each of the solutions to solve for .

Step 7

Step 7.1

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

Step 7.2

Simplify the right side.

Step 7.2.1

The exact value of is .

Step 7.3

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 7.4

Simplify .

Step 7.4.1

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

Step 7.4.2

Combine fractions.

Step 7.4.2.1

Combine and .

Step 7.4.2.2

Combine the numerators over the common denominator.

Step 7.4.3

Simplify the numerator.

Step 7.4.3.1

Multiply by .

Step 7.4.3.2

Subtract from .

Step 7.5

Find the period of .

Step 7.5.1

The period of the function can be calculated using .

Step 7.5.2

Replace with in the formula for period.

Step 7.5.3

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

Step 7.5.4

Divide by .

Step 7.6

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

, for any integer

, for any integer

Step 8

Step 8.1

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

Step 8.2

Simplify the right side.

Step 8.2.1

The exact value of is .

Step 8.3

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 8.4

Simplify .

Step 8.4.1

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

Step 8.4.2

Combine fractions.

Step 8.4.2.1

Combine and .

Step 8.4.2.2

Combine the numerators over the common denominator.

Step 8.4.3

Simplify the numerator.

Step 8.4.3.1

Multiply by .

Step 8.4.3.2

Subtract from .

Step 8.5

Find the period of .

Step 8.5.1

The period of the function can be calculated using .

Step 8.5.2

Replace with in the formula for period.

Step 8.5.3

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

Step 8.5.4

Divide by .

Step 8.6

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

, for any integer

, for any integer

Step 9

List all of the solutions.

, for any integer

Step 10

Step 10.1

Consolidate and to .

, for any integer

Step 10.2

Consolidate and to .

, for any integer

, for any integer