Trigonometry Examples

Find the Inverse (4-cos(10x))^-11
Step 1
Interchange the variables.
Step 2
Solve for .
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Step 2.1
Rewrite the equation as .
Step 2.2
Rewrite the expression using the negative exponent rule .
Step 2.3
Find the LCD of the terms in the equation.
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Step 2.3.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 2.3.2
The LCM of one and any expression is the expression.
Step 2.4
Multiply each term in by to eliminate the fractions.
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Step 2.4.1
Multiply each term in by .
Step 2.4.2
Simplify the left side.
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Step 2.4.2.1
Cancel the common factor of .
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Step 2.4.2.1.1
Cancel the common factor.
Step 2.4.2.1.2
Rewrite the expression.
Step 2.5
Solve the equation.
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Step 2.5.1
Rewrite the equation as .
Step 2.5.2
Divide each term in by and simplify.
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Step 2.5.2.1
Divide each term in by .
Step 2.5.2.2
Simplify the left side.
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Step 2.5.2.2.1
Cancel the common factor of .
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Step 2.5.2.2.1.1
Cancel the common factor.
Step 2.5.2.2.1.2
Divide by .
Step 2.5.3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 2.5.4
Simplify .
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Step 2.5.4.1
Rewrite as .
Step 2.5.4.2
Any root of is .
Step 2.5.4.3
Multiply by .
Step 2.5.4.4
Combine and simplify the denominator.
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Step 2.5.4.4.1
Multiply by .
Step 2.5.4.4.2
Raise to the power of .
Step 2.5.4.4.3
Use the power rule to combine exponents.
Step 2.5.4.4.4
Add and .
Step 2.5.4.4.5
Rewrite as .
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Step 2.5.4.4.5.1
Use to rewrite as .
Step 2.5.4.4.5.2
Apply the power rule and multiply exponents, .
Step 2.5.4.4.5.3
Combine and .
Step 2.5.4.4.5.4
Cancel the common factor of .
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Step 2.5.4.4.5.4.1
Cancel the common factor.
Step 2.5.4.4.5.4.2
Rewrite the expression.
Step 2.5.4.4.5.5
Simplify.
Step 2.5.4.5
Rewrite as .
Step 2.5.5
Subtract from both sides of the equation.
Step 2.5.6
Divide each term in by and simplify.
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Step 2.5.6.1
Divide each term in by .
Step 2.5.6.2
Simplify the left side.
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Step 2.5.6.2.1
Dividing two negative values results in a positive value.
Step 2.5.6.2.2
Divide by .
Step 2.5.6.3
Simplify the right side.
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Step 2.5.6.3.1
Simplify each term.
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Step 2.5.6.3.1.1
Move the negative one from the denominator of .
Step 2.5.6.3.1.2
Rewrite as .
Step 2.5.6.3.1.3
Divide by .
Step 2.6
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 2.7
Divide each term in by and simplify.
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Step 2.7.1
Divide each term in by .
Step 2.7.2
Simplify the left side.
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Step 2.7.2.1
Cancel the common factor of .
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Step 2.7.2.1.1
Cancel the common factor.
Step 2.7.2.1.2
Divide by .
Step 3
Replace with to show the final answer.
Step 4
Verify if is the inverse of .
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Step 4.1
To verify the inverse, check if and .
Step 4.2
Evaluate .
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Step 4.2.1
Set up the composite result function.
Step 4.2.2
Evaluate by substituting in the value of into .
Step 4.2.3
Simplify the numerator.
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Step 4.2.3.1
Multiply the exponents in .
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Step 4.2.3.1.1
Apply the power rule and multiply exponents, .
Step 4.2.3.1.2
Multiply by .
Step 4.2.3.2
Rewrite the expression using the negative exponent rule .
Step 4.2.3.3
Rewrite as .
Step 4.2.3.4
Rewrite as .
Step 4.2.3.5
Rewrite as .
Step 4.2.3.6
Pull terms out from under the radical, assuming real numbers.
Step 4.2.4
Simplify the numerator.
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Step 4.2.4.1
Simplify each term.
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Step 4.2.4.1.1
Move to the numerator using the negative exponent rule .
Step 4.2.4.1.2
Combine and .
Step 4.2.4.1.3
Cancel the common factor of and .
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Step 4.2.4.1.3.1
Factor out of .
Step 4.2.4.1.3.2
Cancel the common factors.
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Step 4.2.4.1.3.2.1
Multiply by .
Step 4.2.4.1.3.2.2
Cancel the common factor.
Step 4.2.4.1.3.2.3
Rewrite the expression.
Step 4.2.4.1.3.2.4
Divide by .
Step 4.2.4.1.4
Apply the distributive property.
Step 4.2.4.1.5
Multiply by .
Step 4.2.4.1.6
Multiply .
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Step 4.2.4.1.6.1
Multiply by .
Step 4.2.4.1.6.2
Multiply by .
Step 4.2.4.2
Combine the opposite terms in .
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Step 4.2.4.2.1
Add and .
Step 4.2.4.2.2
Add and .
Step 4.3
Evaluate .
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Step 4.3.1
Set up the composite result function.
Step 4.3.2
Evaluate by substituting in the value of into .
Step 4.3.3
Simplify each term.
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Step 4.3.3.1
Simplify the numerator.
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Step 4.3.3.1.1
To write as a fraction with a common denominator, multiply by .
Step 4.3.3.1.2
Combine the numerators over the common denominator.
Step 4.3.3.1.3
Simplify the numerator.
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Step 4.3.3.1.3.1
Use to rewrite as .
Step 4.3.3.1.3.2
Factor out of .
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Step 4.3.3.1.3.2.1
Factor out of .
Step 4.3.3.1.3.2.2
Factor out of .
Step 4.3.3.1.3.2.3
Factor out of .
Step 4.3.3.1.4
Move to the denominator using the negative exponent rule .
Step 4.3.3.1.5
Multiply by by adding the exponents.
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Step 4.3.3.1.5.1
Multiply by .
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Step 4.3.3.1.5.1.1
Raise to the power of .
Step 4.3.3.1.5.1.2
Use the power rule to combine exponents.
Step 4.3.3.1.5.2
Write as a fraction with a common denominator.
Step 4.3.3.1.5.3
Combine the numerators over the common denominator.
Step 4.3.3.1.5.4
Subtract from .
Step 4.3.3.2
Cancel the common factor of .
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Step 4.3.3.2.1
Cancel the common factor.
Step 4.3.3.2.2
Rewrite the expression.
Step 4.3.3.3
The functions cosine and arccosine are inverses.
Step 4.3.4
To write as a fraction with a common denominator, multiply by .
Step 4.3.5
Combine the numerators over the common denominator.
Step 4.3.6
Simplify the numerator.
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Step 4.3.6.1
Apply the distributive property.
Step 4.3.6.2
Multiply by .
Step 4.3.6.3
Multiply by .
Step 4.3.6.4
Subtract from .
Step 4.3.6.5
Add and .
Step 4.3.7
Change the sign of the exponent by rewriting the base as its reciprocal.
Step 4.3.8
Multiply the exponents in .
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Step 4.3.8.1
Apply the power rule and multiply exponents, .
Step 4.3.8.2
Cancel the common factor of .
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Step 4.3.8.2.1
Cancel the common factor.
Step 4.3.8.2.2
Rewrite the expression.
Step 4.4
Since and , then is the inverse of .