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Trigonometry Examples
Step 1
Step 1.1
Use the quotient property of logarithms, .
Step 1.2
Multiply by .
Step 1.3
Multiply by .
Step 1.4
Expand the denominator using the FOIL method.
Step 1.5
Simplify.
Step 2
Rewrite in exponential form using the definition of a logarithm. If and are positive real numbers and , then is equivalent to .
Step 3
Cross multiply to remove the fraction.
Step 4
Step 4.1
Raise to the power of .
Step 4.2
Apply the distributive property.
Step 4.3
Multiply.
Step 4.3.1
Multiply by .
Step 4.3.2
Multiply by .
Step 5
Step 5.1
Add to both sides of the equation.
Step 5.2
Simplify each term.
Step 5.2.1
Apply the distributive property.
Step 5.2.2
Multiply by .
Step 5.2.3
Multiply by .
Step 5.3
Add and .
Step 6
Step 6.1
Factor out of .
Step 6.2
Factor out of .
Step 6.3
Factor out of .
Step 7
Step 7.1
Divide each term in by .
Step 7.2
Simplify the left side.
Step 7.2.1
Cancel the common factor of .
Step 7.2.1.1
Cancel the common factor.
Step 7.2.1.2
Divide by .
Step 7.2.2
Apply the distributive property.
Step 7.2.3
Reorder.
Step 7.2.3.1
Rewrite using the commutative property of multiplication.
Step 7.2.3.2
Move to the left of .
Step 7.3
Simplify the right side.
Step 7.3.1
Divide by .
Step 8
Subtract from both sides of the equation.
Step 9
To remove the radical on the left side of the equation, square both sides of the equation.
Step 10
Step 10.1
Use to rewrite as .
Step 10.2
Simplify the left side.
Step 10.2.1
Simplify .
Step 10.2.1.1
Multiply by by adding the exponents.
Step 10.2.1.1.1
Move .
Step 10.2.1.1.2
Multiply by .
Step 10.2.1.1.2.1
Raise to the power of .
Step 10.2.1.1.2.2
Use the power rule to combine exponents.
Step 10.2.1.1.3
Write as a fraction with a common denominator.
Step 10.2.1.1.4
Combine the numerators over the common denominator.
Step 10.2.1.1.5
Add and .
Step 10.2.1.2
Apply the product rule to .
Step 10.2.1.3
Raise to the power of .
Step 10.2.1.4
Multiply the exponents in .
Step 10.2.1.4.1
Apply the power rule and multiply exponents, .
Step 10.2.1.4.2
Cancel the common factor of .
Step 10.2.1.4.2.1
Cancel the common factor.
Step 10.2.1.4.2.2
Rewrite the expression.
Step 10.3
Simplify the right side.
Step 10.3.1
Simplify .
Step 10.3.1.1
Rewrite as .
Step 10.3.1.2
Expand using the FOIL Method.
Step 10.3.1.2.1
Apply the distributive property.
Step 10.3.1.2.2
Apply the distributive property.
Step 10.3.1.2.3
Apply the distributive property.
Step 10.3.1.3
Simplify and combine like terms.
Step 10.3.1.3.1
Simplify each term.
Step 10.3.1.3.1.1
Multiply by .
Step 10.3.1.3.1.2
Multiply by .
Step 10.3.1.3.1.3
Multiply by .
Step 10.3.1.3.1.4
Rewrite using the commutative property of multiplication.
Step 10.3.1.3.1.5
Multiply by by adding the exponents.
Step 10.3.1.3.1.5.1
Move .
Step 10.3.1.3.1.5.2
Multiply by .
Step 10.3.1.3.1.6
Multiply by .
Step 10.3.1.3.2
Subtract from .
Step 11
Step 11.1
Move all the expressions to the left side of the equation.
Step 11.1.1
Subtract from both sides of the equation.
Step 11.1.2
Add to both sides of the equation.
Step 11.1.3
Subtract from both sides of the equation.
Step 11.2
Factor the left side of the equation.
Step 11.2.1
Reorder terms.
Step 11.2.2
Factor using the rational roots test.
Step 11.2.2.1
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 11.2.2.2
Find every combination of . These are the possible roots of the polynomial function.
Step 11.2.2.3
Substitute and simplify the expression. In this case, the expression is equal to so is a root of the polynomial.
Step 11.2.2.3.1
Substitute into the polynomial.
Step 11.2.2.3.2
Raise to the power of .
Step 11.2.2.3.3
Multiply by .
Step 11.2.2.3.4
Raise to the power of .
Step 11.2.2.3.5
Multiply by .
Step 11.2.2.3.6
Subtract from .
Step 11.2.2.3.7
Multiply by .
Step 11.2.2.3.8
Add and .
Step 11.2.2.3.9
Subtract from .
Step 11.2.2.4
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 11.2.2.5
Divide by .
Step 11.2.2.5.1
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 11.2.2.5.2
Divide the highest order term in the dividend by the highest order term in divisor .
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Step 11.2.2.5.3
Multiply the new quotient term by the divisor.
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Step 11.2.2.5.4
The expression needs to be subtracted from the dividend, so change all the signs in
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Step 11.2.2.5.5
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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Step 11.2.2.5.6
Pull the next terms from the original dividend down into the current dividend.
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Step 11.2.2.5.7
Divide the highest order term in the dividend by the highest order term in divisor .
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Step 11.2.2.5.8
Multiply the new quotient term by the divisor.
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Step 11.2.2.5.9
The expression needs to be subtracted from the dividend, so change all the signs in
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Step 11.2.2.5.10
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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Step 11.2.2.5.11
Pull the next terms from the original dividend down into the current dividend.
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Step 11.2.2.5.12
Divide the highest order term in the dividend by the highest order term in divisor .
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Step 11.2.2.5.13
Multiply the new quotient term by the divisor.
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Step 11.2.2.5.14
The expression needs to be subtracted from the dividend, so change all the signs in
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Step 11.2.2.5.15
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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Step 11.2.2.5.16
Since the remander is , the final answer is the quotient.
Step 11.2.2.6
Write as a set of factors.
Step 11.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 11.4
Set equal to and solve for .
Step 11.4.1
Set equal to .
Step 11.4.2
Add to both sides of the equation.
Step 11.5
Set equal to and solve for .
Step 11.5.1
Set equal to .
Step 11.5.2
Solve for .
Step 11.5.2.1
Use the quadratic formula to find the solutions.
Step 11.5.2.2
Substitute the values , , and into the quadratic formula and solve for .
Step 11.5.2.3
Simplify.
Step 11.5.2.3.1
Simplify the numerator.
Step 11.5.2.3.1.1
Raise to the power of .
Step 11.5.2.3.1.2
Multiply .
Step 11.5.2.3.1.2.1
Multiply by .
Step 11.5.2.3.1.2.2
Multiply by .
Step 11.5.2.3.1.3
Subtract from .
Step 11.5.2.3.1.4
Rewrite as .
Step 11.5.2.3.1.4.1
Factor out of .
Step 11.5.2.3.1.4.2
Rewrite as .
Step 11.5.2.3.1.5
Pull terms out from under the radical.
Step 11.5.2.3.2
Multiply by .
Step 11.5.2.4
The final answer is the combination of both solutions.
Step 11.6
The final solution is all the values that make true.
Step 12
Exclude the solutions that do not make true.
Step 13
The result can be shown in multiple forms.
Exact Form:
Decimal Form: