# Trigonometry Examples

Step 1

Step 1.1

Let . Substitute for all occurrences of .

Step 1.2

Factor out of .

Step 1.2.1

Factor out of .

Step 1.2.2

Factor out of .

Step 1.2.3

Factor out of .

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

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

Step 3.2.2

Simplify the right side.

Step 3.2.2.1

The exact value of is .

Step 3.2.3

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.

Step 3.2.4

Subtract from .

Step 3.2.5

Find the period of .

Step 3.2.5.1

The period of the function can be calculated using .

Step 3.2.5.2

Replace with in the formula for period.

Step 3.2.5.3

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

Step 3.2.5.4

Divide by .

Step 3.2.6

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

Add to both sides of the equation.

Step 4.2.2

Divide each term in by and simplify.

Step 4.2.2.1

Divide each term in by .

Step 4.2.2.2

Simplify the left side.

Step 4.2.2.2.1

Cancel the common factor of .

Step 4.2.2.2.1.1

Cancel the common factor.

Step 4.2.2.2.1.2

Divide by .

Step 4.2.3

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

Step 4.2.4

Simplify the right side.

Step 4.2.4.1

The exact value of is .

Step 4.2.5

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.

Step 4.2.6

Simplify .

Step 4.2.6.1

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

Step 4.2.6.2

Combine fractions.

Step 4.2.6.2.1

Combine and .

Step 4.2.6.2.2

Combine the numerators over the common denominator.

Step 4.2.6.3

Simplify the numerator.

Step 4.2.6.3.1

Move to the left of .

Step 4.2.6.3.2

Subtract from .

Step 4.2.7

Find the period of .

Step 4.2.7.1

The period of the function can be calculated using .

Step 4.2.7.2

Replace with in the formula for period.

Step 4.2.7.3

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

Step 4.2.7.4

Divide by .

Step 4.2.8

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 5

The final solution is all the values that make true.

, for any integer

Step 6

Consolidate and to .

, for any integer