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Trigonometry Examples
Step 1
Step 1.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 1.2
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 1.3
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
Step 1.4
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.
Step 1.5
The factor for is itself.
occurs time.
Step 1.6
The factor for is itself.
occurs time.
Step 1.7
The factor for is itself.
occurs time.
Step 1.8
The LCM of is the result of multiplying all factors the greatest number of times they occur in either term.
Step 2
Step 2.1
Multiply each term in by .
Step 2.2
Simplify the left side.
Step 2.2.1
Simplify each term.
Step 2.2.1.1
Cancel the common factor of .
Step 2.2.1.1.1
Factor out of .
Step 2.2.1.1.2
Cancel the common factor.
Step 2.2.1.1.3
Rewrite the expression.
Step 2.2.1.2
Rewrite as .
Step 2.2.1.3
Factor out of .
Step 2.2.1.4
Factor out of .
Step 2.2.1.5
Reorder terms.
Step 2.2.1.6
Raise to the power of .
Step 2.2.1.7
Raise to the power of .
Step 2.2.1.8
Use the power rule to combine exponents.
Step 2.2.1.9
Add and .
Step 2.2.1.10
Rewrite using the commutative property of multiplication.
Step 2.2.1.11
Cancel the common factor of .
Step 2.2.1.11.1
Move the leading negative in into the numerator.
Step 2.2.1.11.2
Factor out of .
Step 2.2.1.11.3
Cancel the common factor.
Step 2.2.1.11.4
Rewrite the expression.
Step 2.2.1.12
Expand using the FOIL Method.
Step 2.2.1.12.1
Apply the distributive property.
Step 2.2.1.12.2
Apply the distributive property.
Step 2.2.1.12.3
Apply the distributive property.
Step 2.2.1.13
Combine the opposite terms in .
Step 2.2.1.13.1
Reorder the factors in the terms and .
Step 2.2.1.13.2
Add and .
Step 2.2.1.13.3
Add and .
Step 2.2.1.14
Simplify each term.
Step 2.2.1.14.1
Multiply by .
Step 2.2.1.14.2
Multiply by .
Step 2.2.1.15
Apply the distributive property.
Step 2.2.1.16
Multiply by .
Step 2.3
Simplify the right side.
Step 2.3.1
Cancel the common factor of .
Step 2.3.1.1
Factor out of .
Step 2.3.1.2
Cancel the common factor.
Step 2.3.1.3
Rewrite the expression.
Step 2.3.2
Expand using the FOIL Method.
Step 2.3.2.1
Apply the distributive property.
Step 2.3.2.2
Apply the distributive property.
Step 2.3.2.3
Apply the distributive property.
Step 2.3.3
Simplify and combine like terms.
Step 2.3.3.1
Simplify each term.
Step 2.3.3.1.1
Move to the left of .
Step 2.3.3.1.2
Rewrite using the commutative property of multiplication.
Step 2.3.3.1.3
Multiply by by adding the exponents.
Step 2.3.3.1.3.1
Move .
Step 2.3.3.1.3.2
Multiply by .
Step 2.3.3.1.4
Multiply by .
Step 2.3.3.1.5
Multiply by .
Step 2.3.3.2
Subtract from .
Step 2.3.3.3
Add and .
Step 2.3.4
Apply the distributive property.
Step 2.3.5
Multiply.
Step 2.3.5.1
Multiply by .
Step 2.3.5.2
Multiply by .
Step 3
Step 3.1
Move all terms containing to the left side of the equation.
Step 3.1.1
Add to both sides of the equation.
Step 3.1.2
Simplify each term.
Step 3.1.2.1
Rewrite as .
Step 3.1.2.2
Expand using the FOIL Method.
Step 3.1.2.2.1
Apply the distributive property.
Step 3.1.2.2.2
Apply the distributive property.
Step 3.1.2.2.3
Apply the distributive property.
Step 3.1.2.3
Simplify and combine like terms.
Step 3.1.2.3.1
Simplify each term.
Step 3.1.2.3.1.1
Multiply by .
Step 3.1.2.3.1.2
Move to the left of .
Step 3.1.2.3.1.3
Multiply by .
Step 3.1.2.3.2
Subtract from .
Step 3.1.2.4
Apply the distributive property.
Step 3.1.2.5
Simplify.
Step 3.1.2.5.1
Multiply by by adding the exponents.
Step 3.1.2.5.1.1
Move .
Step 3.1.2.5.1.2
Multiply by .
Step 3.1.2.5.1.2.1
Raise to the power of .
Step 3.1.2.5.1.2.2
Use the power rule to combine exponents.
Step 3.1.2.5.1.3
Add and .
Step 3.1.2.5.2
Rewrite using the commutative property of multiplication.
Step 3.1.2.5.3
Multiply by .
Step 3.1.2.6
Simplify each term.
Step 3.1.2.6.1
Multiply by by adding the exponents.
Step 3.1.2.6.1.1
Move .
Step 3.1.2.6.1.2
Multiply by .
Step 3.1.2.6.2
Multiply by .
Step 3.1.3
Subtract from .
Step 3.1.4
Add and .
Step 3.2
Subtract from both sides of the equation.
Step 3.3
Subtract from .
Step 3.4
Factor the left side of the equation.
Step 3.4.1
Factor using the rational roots test.
Step 3.4.1.1
If a polynomial function has integer coefficients, then every rational zero will have the form where is a factor of the constant and is a factor of the leading coefficient.
Step 3.4.1.2
Find every combination of . These are the possible roots of the polynomial function.
Step 3.4.1.3
Substitute and simplify the expression. In this case, the expression is equal to so is a root of the polynomial.
Step 3.4.1.3.1
Substitute into the polynomial.
Step 3.4.1.3.2
Raise to the power of .
Step 3.4.1.3.3
Multiply by .
Step 3.4.1.3.4
Raise to the power of .
Step 3.4.1.3.5
Multiply by .
Step 3.4.1.3.6
Add and .
Step 3.4.1.3.7
Multiply by .
Step 3.4.1.3.8
Add and .
Step 3.4.1.3.9
Subtract from .
Step 3.4.1.4
Since is a known root, divide the polynomial by to find the quotient polynomial. This polynomial can then be used to find the remaining roots.
Step 3.4.1.5
Divide by .
Step 3.4.1.5.1
Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of .
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Step 3.4.1.5.2
Divide the highest order term in the dividend by the highest order term in divisor .
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Step 3.4.1.5.3
Multiply the new quotient term by the divisor.
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Step 3.4.1.5.4
The expression needs to be subtracted from the dividend, so change all the signs in
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Step 3.4.1.5.5
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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Step 3.4.1.5.6
Pull the next terms from the original dividend down into the current dividend.
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Step 3.4.1.5.7
Divide the highest order term in the dividend by the highest order term in divisor .
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Step 3.4.1.5.8
Multiply the new quotient term by the divisor.
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Step 3.4.1.5.9
The expression needs to be subtracted from the dividend, so change all the signs in
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Step 3.4.1.5.10
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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Step 3.4.1.5.11
Pull the next terms from the original dividend down into the current dividend.
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Step 3.4.1.5.12
Divide the highest order term in the dividend by the highest order term in divisor .
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Step 3.4.1.5.13
Multiply the new quotient term by the divisor.
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Step 3.4.1.5.14
The expression needs to be subtracted from the dividend, so change all the signs in
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+ | + |
Step 3.4.1.5.15
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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Step 3.4.1.5.16
Since the remander is , the final answer is the quotient.
Step 3.4.1.6
Write as a set of factors.
Step 3.4.2
Factor by grouping.
Step 3.4.2.1
Factor by grouping.
Step 3.4.2.1.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Step 3.4.2.1.1.1
Factor out of .
Step 3.4.2.1.1.2
Rewrite as plus
Step 3.4.2.1.1.3
Apply the distributive property.
Step 3.4.2.1.2
Factor out the greatest common factor from each group.
Step 3.4.2.1.2.1
Group the first two terms and the last two terms.
Step 3.4.2.1.2.2
Factor out the greatest common factor (GCF) from each group.
Step 3.4.2.1.3
Factor the polynomial by factoring out the greatest common factor, .
Step 3.4.2.2
Remove unnecessary parentheses.
Step 3.5
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 3.6
Set equal to and solve for .
Step 3.6.1
Set equal to .
Step 3.6.2
Subtract from both sides of the equation.
Step 3.7
Set equal to and solve for .
Step 3.7.1
Set equal to .
Step 3.7.2
Solve for .
Step 3.7.2.1
Subtract from both sides of the equation.
Step 3.7.2.2
Divide each term in by and simplify.
Step 3.7.2.2.1
Divide each term in by .
Step 3.7.2.2.2
Simplify the left side.
Step 3.7.2.2.2.1
Dividing two negative values results in a positive value.
Step 3.7.2.2.2.2
Divide by .
Step 3.7.2.2.3
Simplify the right side.
Step 3.7.2.2.3.1
Divide by .
Step 3.8
Set equal to and solve for .
Step 3.8.1
Set equal to .
Step 3.8.2
Add to both sides of the equation.
Step 3.9
The final solution is all the values that make true.
Step 4
Exclude the solutions that do not make true.