Trigonometry Examples

Find the Inverse cos(x-y)
Step 1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 2
Simplify the right side.
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Step 2.1
The exact value of is .
Step 3
Subtract from both sides of the equation.
Step 4
Divide each term in by and simplify.
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Step 4.1
Divide each term in by .
Step 4.2
Simplify the left side.
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Step 4.2.1
Dividing two negative values results in a positive value.
Step 4.2.2
Divide by .
Step 4.3
Simplify the right side.
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Step 4.3.1
Simplify each term.
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Step 4.3.1.1
Move the negative one from the denominator of .
Step 4.3.1.2
Rewrite as .
Step 4.3.1.3
Dividing two negative values results in a positive value.
Step 4.3.1.4
Divide by .
Step 5
The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from to find the solution in the fourth quadrant.
Step 6
Solve for .
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Step 6.1
Simplify .
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Step 6.1.1
To write as a fraction with a common denominator, multiply by .
Step 6.1.2
Combine fractions.
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Step 6.1.2.1
Combine and .
Step 6.1.2.2
Combine the numerators over the common denominator.
Step 6.1.3
Simplify the numerator.
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Step 6.1.3.1
Multiply by .
Step 6.1.3.2
Subtract from .
Step 6.2
Subtract from both sides of the equation.
Step 6.3
Divide each term in by and simplify.
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Step 6.3.1
Divide each term in by .
Step 6.3.2
Simplify the left side.
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Step 6.3.2.1
Dividing two negative values results in a positive value.
Step 6.3.2.2
Divide by .
Step 6.3.3
Simplify the right side.
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Step 6.3.3.1
Simplify each term.
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Step 6.3.3.1.1
Move the negative one from the denominator of .
Step 6.3.3.1.2
Rewrite as .
Step 6.3.3.1.3
Dividing two negative values results in a positive value.
Step 6.3.3.1.4
Divide by .
Step 7
Interchange the variables.
Step 8
Solve for .
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Step 8.1
Rewrite the equation as .
Step 8.2
Add to both sides of the equation.
Step 9
Replace with to show the final answer.
Step 10
Verify if is the inverse of .
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Step 10.1
To verify the inverse, check if and .
Step 10.2
Evaluate .
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Step 10.2.1
Set up the composite result function.
Step 10.2.2
Evaluate by substituting in the value of into .
Step 10.2.3
Combine the opposite terms in .
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Step 10.2.3.1
Add and .
Step 10.2.3.2
Add and .
Step 10.3
Evaluate .
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Step 10.3.1
Set up the composite result function.
Step 10.3.2
Evaluate by substituting in the value of into .
Step 10.3.3
Remove parentheses.
Step 10.3.4
Combine the opposite terms in .
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Step 10.3.4.1
Add and .
Step 10.3.4.2
Add and .
Step 10.4
Since and , then is the inverse of .