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Trigonometry Examples
Step 1
Interchange the variables.
Step 2
Step 2.1
Rewrite the equation as .
Step 2.2
Use the identity to solve the equation. In this identity, represents the angle created by plotting point on a graph and therefore can be found using .
where and
Step 2.3
Set up the equation to find the value of .
Step 2.4
Take the inverse tangent to solve the equation for .
Step 2.4.1
Divide by .
Step 2.4.2
The exact value of is .
Step 2.5
Solve to find the value of .
Step 2.5.1
Raise to the power of .
Step 2.5.2
One to any power is one.
Step 2.5.3
Add and .
Step 2.6
Substitute the known values into the equation.
Step 2.7
Divide each term in by and simplify.
Step 2.7.1
Divide each term in by .
Step 2.7.2
Simplify the left side.
Step 2.7.2.1
Cancel the common factor of .
Step 2.7.2.1.1
Cancel the common factor.
Step 2.7.2.1.2
Divide by .
Step 2.7.3
Simplify the right side.
Step 2.7.3.1
Multiply by .
Step 2.7.3.2
Combine and simplify the denominator.
Step 2.7.3.2.1
Multiply by .
Step 2.7.3.2.2
Raise to the power of .
Step 2.7.3.2.3
Raise to the power of .
Step 2.7.3.2.4
Use the power rule to combine exponents.
Step 2.7.3.2.5
Add and .
Step 2.7.3.2.6
Rewrite as .
Step 2.7.3.2.6.1
Use to rewrite as .
Step 2.7.3.2.6.2
Apply the power rule and multiply exponents, .
Step 2.7.3.2.6.3
Combine and .
Step 2.7.3.2.6.4
Cancel the common factor of .
Step 2.7.3.2.6.4.1
Cancel the common factor.
Step 2.7.3.2.6.4.2
Rewrite the expression.
Step 2.7.3.2.6.5
Evaluate the exponent.
Step 2.8
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 2.9
Add to both sides of the equation.
Step 2.10
Divide each term in by and simplify.
Step 2.10.1
Divide each term in by .
Step 2.10.2
Simplify the left side.
Step 2.10.2.1
Cancel the common factor of .
Step 2.10.2.1.1
Cancel the common factor.
Step 2.10.2.1.2
Divide by .
Step 2.10.3
Simplify the right side.
Step 2.10.3.1
Simplify each term.
Step 2.10.3.1.1
Multiply the numerator by the reciprocal of the denominator.
Step 2.10.3.1.2
Multiply .
Step 2.10.3.1.2.1
Multiply by .
Step 2.10.3.1.2.2
Multiply by .
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
Reorder factors in .
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
Simplify each term.
Step 4.3.3.1
Apply the distributive property.
Step 4.3.3.2
Cancel the common factor of .
Step 4.3.3.2.1
Cancel the common factor.
Step 4.3.3.2.2
Rewrite the expression.
Step 4.3.3.3
Cancel the common factor of .
Step 4.3.3.3.1
Factor out of .
Step 4.3.3.3.2
Cancel the common factor.
Step 4.3.3.3.3
Rewrite the expression.
Step 4.3.3.4
Apply the distributive property.
Step 4.3.3.5
Cancel the common factor of .
Step 4.3.3.5.1
Cancel the common factor.
Step 4.3.3.5.2
Rewrite the expression.
Step 4.3.3.6
Cancel the common factor of .
Step 4.3.3.6.1
Factor out of .
Step 4.3.3.6.2
Cancel the common factor.
Step 4.3.3.6.3
Rewrite the expression.
Step 4.4
Since and , then is the inverse of .