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Trigonometry Examples
Step 1
Interchange the variables.
Step 2
Step 2.1
Rewrite the equation as .
Step 2.2
Take the inverse cotangent of both sides of the equation to extract from inside the cotangent.
Step 2.3
Take the inverse arctangent of both sides of the equation to extract from inside the arctangent.
Step 2.4
Simplify the left side.
Step 2.4.1
Simplify .
Step 2.4.1.1
Multiply by .
Step 2.4.1.2
Combine and simplify the denominator.
Step 2.4.1.2.1
Multiply by .
Step 2.4.1.2.2
Raise to the power of .
Step 2.4.1.2.3
Raise to the power of .
Step 2.4.1.2.4
Use the power rule to combine exponents.
Step 2.4.1.2.5
Add and .
Step 2.4.1.2.6
Rewrite as .
Step 2.4.1.2.6.1
Use to rewrite as .
Step 2.4.1.2.6.2
Apply the power rule and multiply exponents, .
Step 2.4.1.2.6.3
Combine and .
Step 2.4.1.2.6.4
Cancel the common factor of .
Step 2.4.1.2.6.4.1
Cancel the common factor.
Step 2.4.1.2.6.4.2
Rewrite the expression.
Step 2.4.1.2.6.5
Evaluate the exponent.
Step 2.5
Simplify the right side.
Step 2.5.1
Draw a triangle in the plane with vertices , , and the origin. Then is the angle between the positive x-axis and the ray beginning at the origin and passing through . Therefore, is .
Step 2.6
Multiply both sides of the equation by .
Step 2.7
Simplify both sides of the equation.
Step 2.7.1
Simplify the left side.
Step 2.7.1.1
Simplify .
Step 2.7.1.1.1
Cancel the common factor of .
Step 2.7.1.1.1.1
Cancel the common factor.
Step 2.7.1.1.1.2
Rewrite the expression.
Step 2.7.1.1.2
Cancel the common factor of .
Step 2.7.1.1.2.1
Factor out of .
Step 2.7.1.1.2.2
Cancel the common factor.
Step 2.7.1.1.2.3
Rewrite the expression.
Step 2.7.2
Simplify the right side.
Step 2.7.2.1
Simplify .
Step 2.7.2.1.1
Multiply by .
Step 2.7.2.1.2
Combine and simplify the denominator.
Step 2.7.2.1.2.1
Multiply by .
Step 2.7.2.1.2.2
Raise to the power of .
Step 2.7.2.1.2.3
Raise to the power of .
Step 2.7.2.1.2.4
Use the power rule to combine exponents.
Step 2.7.2.1.2.5
Add and .
Step 2.7.2.1.2.6
Rewrite as .
Step 2.7.2.1.2.6.1
Use to rewrite as .
Step 2.7.2.1.2.6.2
Apply the power rule and multiply exponents, .
Step 2.7.2.1.2.6.3
Combine and .
Step 2.7.2.1.2.6.4
Cancel the common factor of .
Step 2.7.2.1.2.6.4.1
Cancel the common factor.
Step 2.7.2.1.2.6.4.2
Rewrite the expression.
Step 2.7.2.1.2.6.5
Evaluate the exponent.
Step 2.7.2.1.3
Cancel the common factor of .
Step 2.7.2.1.3.1
Cancel the common factor.
Step 2.7.2.1.3.2
Divide by .
Step 2.7.2.1.4
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
Separate fractions.
Step 4.2.4
Rewrite in terms of sines and cosines.
Step 4.2.5
Multiply by the reciprocal of the fraction to divide by .
Step 4.2.6
Convert from to .
Step 4.2.7
Divide by .
Step 4.2.8
The functions tangent and arctangent are inverses.
Step 4.2.9
Cancel the common factor of .
Step 4.2.9.1
Cancel the common factor.
Step 4.2.9.2
Rewrite the expression.
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
Draw a triangle in the plane with vertices , , and the origin. Then is the angle between the positive x-axis and the ray beginning at the origin and passing through . Therefore, is .
Step 4.3.4
Multiply the numerator by the reciprocal of the denominator.
Step 4.3.5
Multiply by .
Step 4.3.6
Multiply the numerator by the reciprocal of the denominator.
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 .