# Trigonometry Examples

Step 1

The law of sines is based on the proportionality of sides and angles in triangles. The law states that for the angles of a non-right triangle, each angle of the triangle has the same ratio of angle measure to sine value.

Step 2

Substitute the known values into the law of sines to find .

Step 3

Step 3.1

Factor each term.

Step 3.1.1

The exact value of is .

Step 3.1.1.1

Split into two angles where the values of the six trigonometric functions are known.

Step 3.1.1.2

Apply the sum of angles identity.

Step 3.1.1.3

The exact value of is .

Step 3.1.1.4

The exact value of is .

Step 3.1.1.5

The exact value of is .

Step 3.1.1.6

The exact value of is .

Step 3.1.1.7

Simplify .

Step 3.1.1.7.1

Simplify each term.

Step 3.1.1.7.1.1

Multiply .

Step 3.1.1.7.1.1.1

Multiply by .

Step 3.1.1.7.1.1.2

Multiply by .

Step 3.1.1.7.1.2

Multiply .

Step 3.1.1.7.1.2.1

Multiply by .

Step 3.1.1.7.1.2.2

Combine using the product rule for radicals.

Step 3.1.1.7.1.2.3

Multiply by .

Step 3.1.1.7.1.2.4

Multiply by .

Step 3.1.1.7.2

Combine the numerators over the common denominator.

Step 3.1.2

Multiply the numerator by the reciprocal of the denominator.

Step 3.1.3

Multiply by .

Step 3.1.4

Evaluate .

Step 3.1.5

Divide by .

Step 3.2

Find the LCD of the terms in the equation.

Step 3.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

Step 3.2.2

The LCM of one and any expression is the expression.

Step 3.3

Multiply each term in by to eliminate the fractions.

Step 3.3.1

Multiply each term in by .

Step 3.3.2

Simplify the left side.

Step 3.3.2.1

Rewrite using the commutative property of multiplication.

Step 3.3.2.2

Cancel the common factor of .

Step 3.3.2.2.1

Factor out of .

Step 3.3.2.2.2

Cancel the common factor.

Step 3.3.2.2.3

Rewrite the expression.

Step 3.3.2.3

Cancel the common factor of .

Step 3.3.2.3.1

Cancel the common factor.

Step 3.3.2.3.2

Rewrite the expression.

Step 3.3.3

Simplify the right side.

Step 3.3.3.1

Multiply by .

Step 3.4

Solve the equation.

Step 3.4.1

Rewrite the equation as .

Step 3.4.2

Divide each term in by and simplify.

Step 3.4.2.1

Divide each term in by .

Step 3.4.2.2

Simplify the left side.

Step 3.4.2.2.1

Cancel the common factor of .

Step 3.4.2.2.1.1

Cancel the common factor.

Step 3.4.2.2.1.2

Divide by .

Step 3.4.2.3

Simplify the right side.

Step 3.4.2.3.1

Simplify each term.

Step 3.4.2.3.1.1

Evaluate the root.

Step 3.4.2.3.1.2

Divide by .

Step 3.4.2.3.1.3

Evaluate the root.

Step 3.4.2.3.1.4

Divide by .

Step 3.4.2.3.2

Add and .

Step 4

The sum of all the angles in a triangle is degrees.

Step 5

Step 5.1

Add and .

Step 5.2

Move all terms not containing to the right side of the equation.

Step 5.2.1

Subtract from both sides of the equation.

Step 5.2.2

Subtract from .

Step 6

The law of sines is based on the proportionality of sides and angles in triangles. The law states that for the angles of a non-right triangle, each angle of the triangle has the same ratio of angle measure to sine value.

Step 7

Substitute the known values into the law of sines to find .

Step 8

Step 8.1

Factor each term.

Step 8.1.1

Evaluate .

Step 8.1.2

The exact value of is .

Step 8.1.2.1

Split into two angles where the values of the six trigonometric functions are known.

Step 8.1.2.2

Apply the sum of angles identity.

Step 8.1.2.3

The exact value of is .

Step 8.1.2.4

The exact value of is .

Step 8.1.2.5

The exact value of is .

Step 8.1.2.6

The exact value of is .

Step 8.1.2.7

Simplify .

Step 8.1.2.7.1

Simplify each term.

Step 8.1.2.7.1.1

Multiply .

Step 8.1.2.7.1.1.1

Multiply by .

Step 8.1.2.7.1.1.2

Multiply by .

Step 8.1.2.7.1.2

Multiply .

Step 8.1.2.7.1.2.1

Multiply by .

Step 8.1.2.7.1.2.2

Combine using the product rule for radicals.

Step 8.1.2.7.1.2.3

Multiply by .

Step 8.1.2.7.1.2.4

Multiply by .

Step 8.1.2.7.2

Combine the numerators over the common denominator.

Step 8.1.3

Multiply the numerator by the reciprocal of the denominator.

Step 8.1.4

Divide by .

Step 8.1.5

Multiply .

Step 8.1.5.1

Combine and .

Step 8.1.5.2

Multiply by .

Step 8.1.6

Divide by .

Step 8.2

Find the LCD of the terms in the equation.

Step 8.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

Step 8.2.2

The LCM of one and any expression is the expression.

Step 8.3

Multiply each term in by to eliminate the fractions.

Step 8.3.1

Multiply each term in by .

Step 8.3.2

Simplify the left side.

Step 8.3.2.1

Cancel the common factor of .

Step 8.3.2.1.1

Cancel the common factor.

Step 8.3.2.1.2

Rewrite the expression.

Step 8.4

Solve the equation.

Step 8.4.1

Rewrite the equation as .

Step 8.4.2

Divide each term in by and simplify.

Step 8.4.2.1

Divide each term in by .

Step 8.4.2.2

Simplify the left side.

Step 8.4.2.2.1

Cancel the common factor of .

Step 8.4.2.2.1.1

Cancel the common factor.

Step 8.4.2.2.1.2

Divide by .

Step 8.4.2.3

Simplify the right side.

Step 8.4.2.3.1

Divide by .