Trigonometry Examples

Find the Inverse (sin(x+y))/(sin(x-y))
Step 1
Set the numerator equal to zero.
Step 2
Solve the equation for .
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Step 2.1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 2.2
Simplify the right side.
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Step 2.2.1
The exact value of is .
Step 2.3
Subtract from both sides of the equation.
Step 2.4
The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from to find the solution in the second quadrant.
Step 2.5
Solve for .
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Step 2.5.1
Subtract from .
Step 2.5.2
Subtract from both sides of the equation.
Step 2.6
Subtract from both sides of the equation.
Step 3
Interchange the variables.
Step 4
Solve for .
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Step 4.1
Rewrite the equation as .
Step 4.2
Divide each term in by and simplify.
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Step 4.2.1
Divide each term in by .
Step 4.2.2
Simplify the left side.
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Step 4.2.2.1
Dividing two negative values results in a positive value.
Step 4.2.2.2
Divide by .
Step 4.2.3
Simplify the right side.
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Step 4.2.3.1
Move the negative one from the denominator of .
Step 4.2.3.2
Rewrite as .
Step 5
Replace with to show the final answer.
Step 6
Verify if is the inverse of .
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Step 6.1
To verify the inverse, check if and .
Step 6.2
Evaluate .
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Step 6.2.1
Set up the composite result function.
Step 6.2.2
Evaluate by substituting in the value of into .
Step 6.2.3
Multiply .
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Step 6.2.3.1
Multiply by .
Step 6.2.3.2
Multiply by .
Step 6.3
Evaluate .
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Step 6.3.1
Set up the composite result function.
Step 6.3.2
Evaluate by substituting in the value of into .
Step 6.3.3
Multiply .
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Step 6.3.3.1
Multiply by .
Step 6.3.3.2
Multiply by .
Step 6.4
Since and , then is the inverse of .