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Trigonometry Examples
Step 1
Interchange the variables.
Step 2
Step 2.1
Rewrite the equation as .
Step 2.2
Divide each term in by and simplify.
Step 2.2.1
Divide each term in by .
Step 2.2.2
Simplify the left side.
Step 2.2.2.1
Cancel the common factor of .
Step 2.2.2.1.1
Cancel the common factor.
Step 2.2.2.1.2
Divide by .
Step 2.3
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 2.4
Add to both sides of the equation.
Step 2.5
Multiply both sides of the equation by .
Step 2.6
Simplify both sides of the equation.
Step 2.6.1
Simplify the left side.
Step 2.6.1.1
Cancel the common factor of .
Step 2.6.1.1.1
Cancel the common factor.
Step 2.6.1.1.2
Rewrite the expression.
Step 2.6.2
Simplify the right side.
Step 2.6.2.1
Simplify .
Step 2.6.2.1.1
Apply the distributive property.
Step 2.6.2.1.2
Combine and .
Step 2.7
Add to both sides of the equation.
Step 2.8
Multiply both sides of the equation by .
Step 2.9
Simplify both sides of the equation.
Step 2.9.1
Simplify the left side.
Step 2.9.1.1
Cancel the common factor of .
Step 2.9.1.1.1
Cancel the common factor.
Step 2.9.1.1.2
Rewrite the expression.
Step 2.9.2
Simplify the right side.
Step 2.9.2.1
Simplify .
Step 2.9.2.1.1
Apply the distributive property.
Step 2.9.2.1.2
Combine and .
Step 3
Replace with to show the final answer.
Step 4
Step 4.1
To verify the inverse, check if and .
Step 4.2
Evaluate .
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
Cancel the common factor of .
Step 4.2.3.1
Cancel the common factor.
Step 4.2.3.2
Divide by .
Step 4.3
Evaluate .
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
Cancel the common factor of and .
Step 4.3.3.1
Factor out of .
Step 4.3.3.2
Factor out of .
Step 4.3.3.3
Factor out of .
Step 4.3.3.4
Cancel the common factors.
Step 4.3.3.4.1
Factor out of .
Step 4.3.3.4.2
Cancel the common factor.
Step 4.3.3.4.3
Rewrite the expression.
Step 4.3.3.4.4
Divide by .
Step 4.3.4
Combine the opposite terms in .
Step 4.3.4.1
Combine the numerators over the common denominator.
Step 4.3.4.2
Subtract from .
Step 4.3.5
Simplify the expression.
Step 4.3.5.1
Divide by .
Step 4.3.5.2
Add and .
Step 4.3.6
The functions cosine and arccosine are inverses.
Step 4.3.7
Cancel the common factor of .
Step 4.3.7.1
Cancel the common factor.
Step 4.3.7.2
Rewrite the expression.
Step 4.4
Since and , then is the inverse of .