# 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

Evaluate .

Step 3.1.2

The exact value of is .

Step 3.1.2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

Step 3.1.2.2

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

Step 3.1.2.3

Apply the sum of angles identity.

Step 3.1.2.4

The exact value of is .

Step 3.1.2.5

The exact value of is .

Step 3.1.2.6

The exact value of is .

Step 3.1.2.7

The exact value of is .

Step 3.1.2.8

Simplify .

Step 3.1.2.8.1

Simplify each term.

Step 3.1.2.8.1.1

Multiply .

Step 3.1.2.8.1.1.1

Multiply by .

Step 3.1.2.8.1.1.2

Multiply by .

Step 3.1.2.8.1.2

Multiply .

Step 3.1.2.8.1.2.1

Multiply by .

Step 3.1.2.8.1.2.2

Combine using the product rule for radicals.

Step 3.1.2.8.1.2.3

Multiply by .

Step 3.1.2.8.1.2.4

Multiply by .

Step 3.1.2.8.2

Combine the numerators over the common denominator.

Step 3.1.3

Multiply the numerator by the reciprocal of the denominator.

Step 3.1.4

Multiply .

Step 3.1.4.1

Multiply by .

Step 3.1.4.2

Multiply 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

Since contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part then find LCM for the variable part .

Step 3.2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 3.2.4

The number is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 3.2.5

The prime factors for are .

Step 3.2.5.1

has factors of and .

Step 3.2.5.2

has factors of and .

Step 3.2.5.3

has factors of and .

Step 3.2.5.4

has factors of and .

Step 3.2.6

Multiply .

Step 3.2.6.1

Multiply by .

Step 3.2.6.2

Multiply by .

Step 3.2.6.3

Multiply by .

Step 3.2.6.4

Multiply by .

Step 3.2.7

The factor for is itself.

occurs time.

Step 3.2.8

The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either term.

Step 3.2.9

The LCM for is the numeric part multiplied by the variable part.

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

Multiply .

Step 3.3.2.2.1

Combine and .

Step 3.3.2.2.2

Multiply by .

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

Cancel the common factor of .

Step 3.3.3.1.1

Factor out of .

Step 3.3.3.1.2

Cancel the common factor.

Step 3.3.3.1.3

Rewrite the expression.

Step 3.3.3.2

Apply the distributive property.

Step 3.4

Solve the equation.

Step 3.4.1

Rewrite the equation as .

Step 3.4.2

Factor out of .

Step 3.4.2.1

Factor out of .

Step 3.4.2.2

Factor out of .

Step 3.4.2.3

Factor out of .

Step 3.4.3

Divide each term in by and simplify.

Step 3.4.3.1

Divide each term in by .

Step 3.4.3.2

Simplify the left side.

Step 3.4.3.2.1

Cancel the common factor of .

Step 3.4.3.2.1.1

Cancel the common factor.

Step 3.4.3.2.1.2

Divide by .

Step 3.4.3.3

Simplify the right side.

Step 3.4.3.3.1

Multiply by .

Step 3.4.3.3.2

Multiply by .

Step 3.4.3.3.3

Expand the denominator using the FOIL method.

Step 3.4.3.3.4

Simplify.

Step 3.4.3.3.5

Multiply by .

Step 3.4.3.3.6

Divide by .

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

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

Step 8.1.2.2

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

Step 8.1.2.3

Apply the sum of angles identity.

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

The exact value of is .

Step 8.1.2.8

Simplify .

Step 8.1.2.8.1

Simplify each term.

Step 8.1.2.8.1.1

Multiply .

Step 8.1.2.8.1.1.1

Multiply by .

Step 8.1.2.8.1.1.2

Multiply by .

Step 8.1.2.8.1.2

Multiply .

Step 8.1.2.8.1.2.1

Multiply by .

Step 8.1.2.8.1.2.2

Combine using the product rule for radicals.

Step 8.1.2.8.1.2.3

Multiply by .

Step 8.1.2.8.1.2.4

Multiply by .

Step 8.1.2.8.2

Combine the numerators over the common denominator.

Step 8.1.3

Multiply the numerator by the reciprocal of the denominator.

Step 8.1.4

Multiply .

Step 8.1.4.1

Multiply by .

Step 8.1.4.2

Multiply 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

Since contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part then find LCM for the variable part .

Step 8.2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 8.2.4

The number is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 8.2.5

The prime factors for are .

Step 8.2.5.1

has factors of and .

Step 8.2.5.2

has factors of and .

Step 8.2.5.3

has factors of and .

Step 8.2.5.4

has factors of and .

Step 8.2.6

Multiply .

Step 8.2.6.1

Multiply by .

Step 8.2.6.2

Multiply by .

Step 8.2.6.3

Multiply by .

Step 8.2.6.4

Multiply by .

Step 8.2.7

The factor for is itself.

occurs time.

Step 8.2.8

The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either term.

Step 8.2.9

The LCM for is the numeric part multiplied by the variable part.

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

Rewrite using the commutative property of multiplication.

Step 8.3.2.2

Multiply .

Step 8.3.2.2.1

Combine and .

Step 8.3.2.2.2

Multiply by .

Step 8.3.2.3

Cancel the common factor of .

Step 8.3.2.3.1

Cancel the common factor.

Step 8.3.2.3.2

Rewrite the expression.

Step 8.3.3

Simplify the right side.

Step 8.3.3.1

Cancel the common factor of .

Step 8.3.3.1.1

Factor out of .

Step 8.3.3.1.2

Cancel the common factor.

Step 8.3.3.1.3

Rewrite the expression.

Step 8.3.3.2

Apply the distributive property.

Step 8.4

Solve the equation.

Step 8.4.1

Rewrite the equation as .

Step 8.4.2

Factor out of .

Step 8.4.2.1

Factor out of .

Step 8.4.2.2

Factor out of .

Step 8.4.2.3

Factor out of .

Step 8.4.3

Divide each term in by and simplify.

Step 8.4.3.1

Divide each term in by .

Step 8.4.3.2

Simplify the left side.

Step 8.4.3.2.1

Cancel the common factor of .

Step 8.4.3.2.1.1

Cancel the common factor.

Step 8.4.3.2.1.2

Divide by .

Step 8.4.3.3

Simplify the right side.

Step 8.4.3.3.1

Multiply by .

Step 8.4.3.3.2

Multiply by .

Step 8.4.3.3.3

Expand the denominator using the FOIL method.

Step 8.4.3.3.4

Simplify.

Step 8.4.3.3.5

Multiply by .

Step 8.4.3.3.6

Divide by .

Step 9

These are the results for all angles and sides for the given triangle.