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Trigonometry Examples
Step 1
Step 1.1
Combine and .
Step 1.2
Multiply by .
Step 2
Step 2.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 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 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 2.4
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
Step 2.5
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.
Step 2.6
The factors for are , which is multiplied by each other times.
occurs times.
Step 2.7
The factor for is itself.
occurs time.
Step 2.8
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either term.
Step 2.9
Simplify .
Step 2.9.1
Multiply by .
Step 2.9.2
Multiply by by adding the exponents.
Step 2.9.2.1
Multiply by .
Step 2.9.2.1.1
Raise to the power of .
Step 2.9.2.1.2
Use the power rule to combine exponents.
Step 2.9.2.2
Add and .
Step 2.9.3
Multiply by by adding the exponents.
Step 2.9.3.1
Multiply by .
Step 2.9.3.1.1
Raise to the power of .
Step 2.9.3.1.2
Use the power rule to combine exponents.
Step 2.9.3.2
Add and .
Step 2.9.4
Multiply by by adding the exponents.
Step 2.9.4.1
Multiply by .
Step 2.9.4.1.1
Raise to the power of .
Step 2.9.4.1.2
Use the power rule to combine exponents.
Step 2.9.4.2
Add and .
Step 2.9.5
Multiply by by adding the exponents.
Step 2.9.5.1
Multiply by .
Step 2.9.5.1.1
Raise to the power of .
Step 2.9.5.1.2
Use the power rule to combine exponents.
Step 2.9.5.2
Add and .
Step 2.9.6
Multiply by by adding the exponents.
Step 2.9.6.1
Multiply by .
Step 2.9.6.1.1
Raise to the power of .
Step 2.9.6.1.2
Use the power rule to combine exponents.
Step 2.9.6.2
Add and .
Step 2.9.7
Multiply by by adding the exponents.
Step 2.9.7.1
Multiply by .
Step 2.9.7.1.1
Raise to the power of .
Step 2.9.7.1.2
Use the power rule to combine exponents.
Step 2.9.7.2
Add and .
Step 2.9.8
Multiply by by adding the exponents.
Step 2.9.8.1
Multiply by .
Step 2.9.8.1.1
Raise to the power of .
Step 2.9.8.1.2
Use the power rule to combine exponents.
Step 2.9.8.2
Add and .
Step 2.9.9
Multiply by by adding the exponents.
Step 2.9.9.1
Multiply by .
Step 2.9.9.1.1
Raise to the power of .
Step 2.9.9.1.2
Use the power rule to combine exponents.
Step 2.9.9.2
Add and .
Step 2.9.10
Multiply by by adding the exponents.
Step 2.9.10.1
Multiply by .
Step 2.9.10.1.1
Raise to the power of .
Step 2.9.10.1.2
Use the power rule to combine exponents.
Step 2.9.10.2
Add and .
Step 2.9.11
Multiply by by adding the exponents.
Step 2.9.11.1
Multiply by .
Step 2.9.11.1.1
Raise to the power of .
Step 2.9.11.1.2
Use the power rule to combine exponents.
Step 2.9.11.2
Add and .
Step 2.9.12
Multiply by by adding the exponents.
Step 2.9.12.1
Multiply by .
Step 2.9.12.1.1
Raise to the power of .
Step 2.9.12.1.2
Use the power rule to combine exponents.
Step 2.9.12.2
Add and .
Step 2.9.13
Multiply by by adding the exponents.
Step 2.9.13.1
Multiply by .
Step 2.9.13.1.1
Raise to the power of .
Step 2.9.13.1.2
Use the power rule to combine exponents.
Step 2.9.13.2
Add and .
Step 2.9.14
Multiply by by adding the exponents.
Step 2.9.14.1
Multiply by .
Step 2.9.14.1.1
Raise to the power of .
Step 2.9.14.1.2
Use the power rule to combine exponents.
Step 2.9.14.2
Add and .
Step 2.9.15
Multiply by by adding the exponents.
Step 2.9.15.1
Multiply by .
Step 2.9.15.1.1
Raise to the power of .
Step 2.9.15.1.2
Use the power rule to combine exponents.
Step 2.9.15.2
Add and .
Step 2.9.16
Multiply by by adding the exponents.
Step 2.9.16.1
Multiply by .
Step 2.9.16.1.1
Raise to the power of .
Step 2.9.16.1.2
Use the power rule to combine exponents.
Step 2.9.16.2
Add and .
Step 2.9.17
Multiply by by adding the exponents.
Step 2.9.17.1
Multiply by .
Step 2.9.17.1.1
Raise to the power of .
Step 2.9.17.1.2
Use the power rule to combine exponents.
Step 2.9.17.2
Add and .
Step 2.9.18
Multiply by by adding the exponents.
Step 2.9.18.1
Multiply by .
Step 2.9.18.1.1
Raise to the power of .
Step 2.9.18.1.2
Use the power rule to combine exponents.
Step 2.9.18.2
Add and .
Step 2.9.19
Multiply by by adding the exponents.
Step 2.9.19.1
Multiply by .
Step 2.9.19.1.1
Raise to the power of .
Step 2.9.19.1.2
Use the power rule to combine exponents.
Step 2.9.19.2
Add and .
Step 2.9.20
Multiply by by adding the exponents.
Step 2.9.20.1
Multiply by .
Step 2.9.20.1.1
Raise to the power of .
Step 2.9.20.1.2
Use the power rule to combine exponents.
Step 2.9.20.2
Add and .
Step 2.9.21
Multiply by by adding the exponents.
Step 2.9.21.1
Multiply by .
Step 2.9.21.1.1
Raise to the power of .
Step 2.9.21.1.2
Use the power rule to combine exponents.
Step 2.9.21.2
Add and .
Step 2.9.22
Multiply by by adding the exponents.
Step 2.9.22.1
Multiply by .
Step 2.9.22.1.1
Raise to the power of .
Step 2.9.22.1.2
Use the power rule to combine exponents.
Step 2.9.22.2
Add and .
Step 2.9.23
Multiply by by adding the exponents.
Step 2.9.23.1
Multiply by .
Step 2.9.23.1.1
Raise to the power of .
Step 2.9.23.1.2
Use the power rule to combine exponents.
Step 2.9.23.2
Add and .
Step 2.9.24
Multiply by by adding the exponents.
Step 2.9.24.1
Multiply by .
Step 2.9.24.1.1
Raise to the power of .
Step 2.9.24.1.2
Use the power rule to combine exponents.
Step 2.9.24.2
Add and .
Step 2.9.25
Multiply by by adding the exponents.
Step 2.9.25.1
Multiply by .
Step 2.9.25.1.1
Raise to the power of .
Step 2.9.25.1.2
Use the power rule to combine exponents.
Step 2.9.25.2
Add and .
Step 2.9.26
Multiply by by adding the exponents.
Step 2.9.26.1
Multiply by .
Step 2.9.26.1.1
Raise to the power of .
Step 2.9.26.1.2
Use the power rule to combine exponents.
Step 2.9.26.2
Add and .
Step 2.9.27
Multiply by by adding the exponents.
Step 2.9.27.1
Multiply by .
Step 2.9.27.1.1
Raise to the power of .
Step 2.9.27.1.2
Use the power rule to combine exponents.
Step 2.9.27.2
Add and .
Step 2.9.28
Multiply by by adding the exponents.
Step 2.9.28.1
Multiply by .
Step 2.9.28.1.1
Raise to the power of .
Step 2.9.28.1.2
Use the power rule to combine exponents.
Step 2.9.28.2
Add and .
Step 2.9.29
Multiply by by adding the exponents.
Step 2.9.29.1
Multiply by .
Step 2.9.29.1.1
Raise to the power of .
Step 2.9.29.1.2
Use the power rule to combine exponents.
Step 2.9.29.2
Add and .
Step 2.9.30
Multiply by by adding the exponents.
Step 2.9.30.1
Multiply by .
Step 2.9.30.1.1
Raise to the power of .
Step 2.9.30.1.2
Use the power rule to combine exponents.
Step 2.9.30.2
Add and .
Step 2.9.31
Multiply by by adding the exponents.
Step 2.9.31.1
Multiply by .
Step 2.9.31.1.1
Raise to the power of .
Step 2.9.31.1.2
Use the power rule to combine exponents.
Step 2.9.31.2
Add and .
Step 2.9.32
Multiply by by adding the exponents.
Step 2.9.32.1
Multiply by .
Step 2.9.32.1.1
Raise to the power of .
Step 2.9.32.1.2
Use the power rule to combine exponents.
Step 2.9.32.2
Add and .
Step 2.9.33
Multiply by by adding the exponents.
Step 2.9.33.1
Multiply by .
Step 2.9.33.1.1
Raise to the power of .
Step 2.9.33.1.2
Use the power rule to combine exponents.
Step 2.9.33.2
Add and .
Step 2.9.34
Multiply by by adding the exponents.
Step 2.9.34.1
Multiply by .
Step 2.9.34.1.1
Raise to the power of .
Step 2.9.34.1.2
Use the power rule to combine exponents.
Step 2.9.34.2
Add and .
Step 2.9.35
Multiply by by adding the exponents.
Step 2.9.35.1
Multiply by .
Step 2.9.35.1.1
Raise to the power of .
Step 2.9.35.1.2
Use the power rule to combine exponents.
Step 2.9.35.2
Add and .
Step 2.9.36
Multiply by by adding the exponents.
Step 2.9.36.1
Multiply by .
Step 2.9.36.1.1
Raise to the power of .
Step 2.9.36.1.2
Use the power rule to combine exponents.
Step 2.9.36.2
Add and .
Step 2.9.37
Multiply by by adding the exponents.
Step 2.9.37.1
Multiply by .
Step 2.9.37.1.1
Raise to the power of .
Step 2.9.37.1.2
Use the power rule to combine exponents.
Step 2.9.37.2
Add and .
Step 2.9.38
Multiply by by adding the exponents.
Step 2.9.38.1
Multiply by .
Step 2.9.38.1.1
Raise to the power of .
Step 2.9.38.1.2
Use the power rule to combine exponents.
Step 2.9.38.2
Add and .
Step 2.9.39
Multiply by by adding the exponents.
Step 2.9.39.1
Multiply by .
Step 2.9.39.1.1
Raise to the power of .
Step 2.9.39.1.2
Use the power rule to combine exponents.
Step 2.9.39.2
Add and .
Step 2.9.40
Multiply by by adding the exponents.
Step 2.9.40.1
Multiply by .
Step 2.9.40.1.1
Raise to the power of .
Step 2.9.40.1.2
Use the power rule to combine exponents.
Step 2.9.40.2
Add and .
Step 2.9.41
Multiply by by adding the exponents.
Step 2.9.41.1
Multiply by .
Step 2.9.41.1.1
Raise to the power of .
Step 2.9.41.1.2
Use the power rule to combine exponents.
Step 2.9.41.2
Add and .
Step 2.9.42
Multiply by by adding the exponents.
Step 2.9.42.1
Multiply by .
Step 2.9.42.1.1
Raise to the power of .
Step 2.9.42.1.2
Use the power rule to combine exponents.
Step 2.9.42.2
Add and .
Step 2.9.43
Multiply by by adding the exponents.
Step 2.9.43.1
Multiply by .
Step 2.9.43.1.1
Raise to the power of .
Step 2.9.43.1.2
Use the power rule to combine exponents.
Step 2.9.43.2
Add and .
Step 2.9.44
Multiply by by adding the exponents.
Step 2.9.44.1
Multiply by .
Step 2.9.44.1.1
Raise to the power of .
Step 2.9.44.1.2
Use the power rule to combine exponents.
Step 2.9.44.2
Add and .
Step 2.9.45
Multiply by by adding the exponents.
Step 2.9.45.1
Multiply by .
Step 2.9.45.1.1
Raise to the power of .
Step 2.9.45.1.2
Use the power rule to combine exponents.
Step 2.9.45.2
Add and .
Step 2.9.46
Multiply by by adding the exponents.
Step 2.9.46.1
Multiply by .
Step 2.9.46.1.1
Raise to the power of .
Step 2.9.46.1.2
Use the power rule to combine exponents.
Step 2.9.46.2
Add and .
Step 2.9.47
Multiply by by adding the exponents.
Step 2.9.47.1
Multiply by .
Step 2.9.47.1.1
Raise to the power of .
Step 2.9.47.1.2
Use the power rule to combine exponents.
Step 2.9.47.2
Add and .
Step 2.9.48
Multiply by by adding the exponents.
Step 2.9.48.1
Multiply by .
Step 2.9.48.1.1
Raise to the power of .
Step 2.9.48.1.2
Use the power rule to combine exponents.
Step 2.9.48.2
Add and .
Step 2.9.49
Multiply by by adding the exponents.
Step 2.9.49.1
Multiply by .
Step 2.9.49.1.1
Raise to the power of .
Step 2.9.49.1.2
Use the power rule to combine exponents.
Step 2.9.49.2
Add and .
Step 2.9.50
Multiply by by adding the exponents.
Step 2.9.50.1
Multiply by .
Step 2.9.50.1.1
Raise to the power of .
Step 2.9.50.1.2
Use the power rule to combine exponents.
Step 2.9.50.2
Add and .
Step 2.9.51
Multiply by by adding the exponents.
Step 2.9.51.1
Multiply by .
Step 2.9.51.1.1
Raise to the power of .
Step 2.9.51.1.2
Use the power rule to combine exponents.
Step 2.9.51.2
Add and .
Step 2.9.52
Multiply by by adding the exponents.
Step 2.9.52.1
Multiply by .
Step 2.9.52.1.1
Raise to the power of .
Step 2.9.52.1.2
Use the power rule to combine exponents.
Step 2.9.52.2
Add and .
Step 2.9.53
Multiply by by adding the exponents.
Step 2.9.53.1
Multiply by .
Step 2.9.53.1.1
Raise to the power of .
Step 2.9.53.1.2
Use the power rule to combine exponents.
Step 2.9.53.2
Add and .
Step 2.9.54
Multiply by by adding the exponents.
Step 2.9.54.1
Multiply by .
Step 2.9.54.1.1
Raise to the power of .
Step 2.9.54.1.2
Use the power rule to combine exponents.
Step 2.9.54.2
Add and .
Step 2.9.55
Multiply by by adding the exponents.
Step 2.9.55.1
Multiply by .
Step 2.9.55.1.1
Raise to the power of .
Step 2.9.55.1.2
Use the power rule to combine exponents.
Step 2.9.55.2
Add and .
Step 2.9.56
Multiply by by adding the exponents.
Step 2.9.56.1
Multiply by .
Step 2.9.56.1.1
Raise to the power of .
Step 2.9.56.1.2
Use the power rule to combine exponents.
Step 2.9.56.2
Add and .
Step 2.9.57
Multiply by by adding the exponents.
Step 2.9.57.1
Multiply by .
Step 2.9.57.1.1
Raise to the power of .
Step 2.9.57.1.2
Use the power rule to combine exponents.
Step 2.9.57.2
Add and .
Step 3
Step 3.1
Multiply each term in by .
Step 3.2
Simplify the left side.
Step 3.2.1
Cancel the common factor of .
Step 3.2.1.1
Factor out of .
Step 3.2.1.2
Cancel the common factor.
Step 3.2.1.3
Rewrite the expression.
Step 3.3
Simplify the right side.
Step 3.3.1
Cancel the common factor of .
Step 3.3.1.1
Factor out of .
Step 3.3.1.2
Cancel the common factor.
Step 3.3.1.3
Rewrite the expression.
Step 4
Step 4.1
Subtract from both sides of the equation.
Step 4.2
Factor out of .
Step 4.2.1
Factor out of .
Step 4.2.2
Factor out of .
Step 4.2.3
Factor out of .
Step 4.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 4.4
Set equal to and solve for .
Step 4.4.1
Set equal to .
Step 4.4.2
Solve for .
Step 4.4.2.1
Convert the decimal exponent to a fractional exponent.
Step 4.4.2.1.1
Convert the decimal number to a fraction by placing the decimal number over a power of ten. Since there are numbers to the right of the decimal point, place the decimal number over . Next, add the whole number to the left of the decimal.
Step 4.4.2.1.2
Reduce the fraction.
Step 4.4.2.1.2.1
Convert to an improper fraction.
Step 4.4.2.1.2.1.1
A mixed number is an addition of its whole and fractional parts.
Step 4.4.2.1.2.1.2
Add and .
Step 4.4.2.1.2.2
Cancel the common factor of and .
Step 4.4.2.1.2.2.1
Factor out of .
Step 4.4.2.1.2.2.2
Cancel the common factors.
Step 4.4.2.1.2.2.2.1
Factor out of .
Step 4.4.2.1.2.2.2.2
Cancel the common factor.
Step 4.4.2.1.2.2.2.3
Rewrite the expression.
Step 4.4.2.2
Raise each side of the equation to the power of to eliminate the fractional exponent on the left side.
Step 4.4.2.3
Simplify the exponent.
Step 4.4.2.3.1
Simplify the left side.
Step 4.4.2.3.1.1
Simplify .
Step 4.4.2.3.1.1.1
Multiply the exponents in .
Step 4.4.2.3.1.1.1.1
Apply the power rule and multiply exponents, .
Step 4.4.2.3.1.1.1.2
Cancel the common factor of .
Step 4.4.2.3.1.1.1.2.1
Factor out of .
Step 4.4.2.3.1.1.1.2.2
Cancel the common factor.
Step 4.4.2.3.1.1.1.2.3
Rewrite the expression.
Step 4.4.2.3.1.1.1.3
Divide by .
Step 4.4.2.3.1.1.2
Simplify.
Step 4.4.2.3.2
Simplify the right side.
Step 4.4.2.3.2.1
Simplify .
Step 4.4.2.3.2.1.1
Divide by .
Step 4.4.2.3.2.1.2
Raising to any positive power yields .
Step 4.5
Set equal to and solve for .
Step 4.5.1
Set equal to .
Step 4.5.2
Solve for .
Step 4.5.2.1
Subtract from both sides of the equation.
Step 4.5.2.2
Convert the decimal exponent to a fractional exponent.
Step 4.5.2.2.1
Convert the decimal number to a fraction by placing the decimal number over a power of ten. Since there are numbers to the right of the decimal point, place the decimal number over . Next, add the whole number to the left of the decimal.
Step 4.5.2.2.2
Reduce the fractional part of the mixed number.
Step 4.5.2.2.3
Convert to an improper fraction.
Step 4.5.2.2.3.1
A mixed number is an addition of its whole and fractional parts.
Step 4.5.2.2.3.2
Add and .
Step 4.5.2.2.3.2.1
To write as a fraction with a common denominator, multiply by .
Step 4.5.2.2.3.2.2
Combine and .
Step 4.5.2.2.3.2.3
Combine the numerators over the common denominator.
Step 4.5.2.2.3.2.4
Simplify the numerator.
Step 4.5.2.2.3.2.4.1
Multiply by .
Step 4.5.2.2.3.2.4.2
Add and .
Step 4.5.2.3
Raise each side of the equation to the power of to eliminate the fractional exponent on the left side.
Step 4.5.2.4
Simplify the exponent.
Step 4.5.2.4.1
Simplify the left side.
Step 4.5.2.4.1.1
Simplify .
Step 4.5.2.4.1.1.1
Divide by .
Step 4.5.2.4.1.1.2
Apply the product rule to .
Step 4.5.2.4.1.1.3
Multiply the exponents in .
Step 4.5.2.4.1.1.3.1
Apply the power rule and multiply exponents, .
Step 4.5.2.4.1.1.3.2
Multiply .
Step 4.5.2.4.1.1.3.2.1
Combine and .
Step 4.5.2.4.1.1.3.2.2
Multiply by .
Step 4.5.2.4.1.1.3.3
Divide by .
Step 4.5.2.4.2
Simplify the right side.
Step 4.5.2.4.2.1
Divide by .
Step 4.5.2.5
Solve for .
Step 4.5.2.5.1
Divide each term in by and simplify.
Step 4.5.2.5.1.1
Divide each term in by .
Step 4.5.2.5.1.2
Simplify the left side.
Step 4.5.2.5.1.2.1
Cancel the common factor of .
Step 4.5.2.5.1.2.1.1
Cancel the common factor.
Step 4.5.2.5.1.2.1.2
Divide by .
Step 4.5.2.5.1.3
Simplify the right side.
Step 4.5.2.5.1.3.1
Cancel the common factor of and .
Step 4.5.2.5.1.3.1.1
Cancel the common factor.
Step 4.5.2.5.1.3.1.2
Rewrite the expression.
Step 4.5.2.5.2
Raise each side of the equation to the power of to eliminate the fractional exponent on the left side.
Step 4.5.2.5.3
Simplify the exponent.
Step 4.5.2.5.3.1
Simplify the left side.
Step 4.5.2.5.3.1.1
Simplify .
Step 4.5.2.5.3.1.1.1
Multiply the exponents in .
Step 4.5.2.5.3.1.1.1.1
Apply the power rule and multiply exponents, .
Step 4.5.2.5.3.1.1.1.2
Cancel the common factor of .
Step 4.5.2.5.3.1.1.1.2.1
Cancel the common factor.
Step 4.5.2.5.3.1.1.1.2.2
Rewrite the expression.
Step 4.5.2.5.3.1.1.2
Simplify.
Step 4.5.2.5.3.2
Simplify the right side.
Step 4.5.2.5.3.2.1
Simplify .
Step 4.5.2.5.3.2.1.1
Divide by .
Step 4.5.2.5.3.2.1.2
One to any power is one.
Step 4.6
The final solution is all the values that make true.
Step 5
Exclude the solutions that do not make true.