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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
Rewrite as .
Step 2.4.1.2
Multiply by .
Step 2.4.1.3
Combine and simplify the denominator.
Step 2.4.1.3.1
Multiply by .
Step 2.4.1.3.2
Raise to the power of .
Step 2.4.1.3.3
Raise to the power of .
Step 2.4.1.3.4
Use the power rule to combine exponents.
Step 2.4.1.3.5
Add and .
Step 2.4.1.3.6
Rewrite as .
Step 2.4.1.3.6.1
Use to rewrite as .
Step 2.4.1.3.6.2
Apply the power rule and multiply exponents, .
Step 2.4.1.3.6.3
Combine and .
Step 2.4.1.3.6.4
Cancel the common factor of .
Step 2.4.1.3.6.4.1
Cancel the common factor.
Step 2.4.1.3.6.4.2
Rewrite the expression.
Step 2.4.1.3.6.5
Simplify.
Step 2.4.1.4
Combine using the product rule for radicals.
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
Cross multiply.
Step 2.6.1
Cross multiply by setting the product of the numerator of the right side and the denominator of the left side equal to the product of the numerator of the left side and the denominator of the right side.
Step 2.6.2
Simplify the left side.
Step 2.6.2.1
Multiply by .
Step 2.6.3
Simplify the right side.
Step 2.6.3.1
Multiply by .
Step 2.7
Rewrite the equation as .
Step 2.8
To remove the radical on the left side of the equation, square both sides of the equation.
Step 2.9
Simplify each side of the equation.
Step 2.9.1
Use to rewrite as .
Step 2.9.2
Simplify the left side.
Step 2.9.2.1
Simplify .
Step 2.9.2.1.1
Apply the product rule to .
Step 2.9.2.1.2
Use the power rule to distribute the exponent.
Step 2.9.2.1.2.1
Apply the product rule to .
Step 2.9.2.1.2.2
Apply the product rule to .
Step 2.9.2.1.3
Multiply the exponents in .
Step 2.9.2.1.3.1
Apply the power rule and multiply exponents, .
Step 2.9.2.1.3.2
Cancel the common factor of .
Step 2.9.2.1.3.2.1
Cancel the common factor.
Step 2.9.2.1.3.2.2
Rewrite the expression.
Step 2.9.2.1.4
Evaluate the exponent.
Step 2.9.2.1.5
Multiply the exponents in .
Step 2.9.2.1.5.1
Apply the power rule and multiply exponents, .
Step 2.9.2.1.5.2
Cancel the common factor of .
Step 2.9.2.1.5.2.1
Cancel the common factor.
Step 2.9.2.1.5.2.2
Rewrite the expression.
Step 2.9.2.1.6
Simplify.
Step 2.10
Solve for .
Step 2.10.1
Subtract from both sides of the equation.
Step 2.10.2
Factor out of .
Step 2.10.2.1
Factor out of .
Step 2.10.2.2
Factor out of .
Step 2.10.2.3
Factor out of .
Step 2.10.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 2.10.4
Set equal to .
Step 2.10.5
Set equal to and solve for .
Step 2.10.5.1
Set equal to .
Step 2.10.5.2
Solve for .
Step 2.10.5.2.1
Subtract from both sides of the equation.
Step 2.10.5.2.2
Divide each term in by and simplify.
Step 2.10.5.2.2.1
Divide each term in by .
Step 2.10.5.2.2.2
Simplify the left side.
Step 2.10.5.2.2.2.1
Dividing two negative values results in a positive value.
Step 2.10.5.2.2.2.2
Divide by .
Step 2.10.5.2.2.3
Simplify the right side.
Step 2.10.5.2.2.3.1
Move the negative one from the denominator of .
Step 2.10.5.2.2.3.2
Rewrite as .
Step 2.10.5.2.2.3.3
Multiply by .
Step 2.10.6
The final solution is all the values that make true.
Step 3
Replace with to show the final answer.
Step 4
Step 4.1
The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of and and compare them.
Step 4.2
Find the domain of .
Step 4.2.1
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
Step 4.3
Since the domain of is not equal to the range of , then is not an inverse of .
There is no inverse
There is no inverse
Step 5