Trigonometry Examples

Find the Inverse y=arcsin(5-3x^2)
Step 1
Interchange the variables.
Step 2
Solve for .
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Step 2.1
Rewrite the equation as .
Step 2.2
Take the inverse arcsine of both sides of the equation to extract from inside the arcsine.
Step 2.3
Subtract from both sides of the equation.
Step 2.4
Divide each term in by and simplify.
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Step 2.4.1
Divide each term in by .
Step 2.4.2
Simplify the left side.
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Step 2.4.2.1
Cancel the common factor of .
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Step 2.4.2.1.1
Cancel the common factor.
Step 2.4.2.1.2
Divide by .
Step 2.4.3
Simplify the right side.
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Step 2.4.3.1
Simplify each term.
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Step 2.4.3.1.1
Move the negative in front of the fraction.
Step 2.4.3.1.2
Dividing two negative values results in a positive value.
Step 2.5
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 2.6
The complete solution is the result of both the positive and negative portions of the solution.
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Step 2.6.1
First, use the positive value of the to find the first solution.
Step 2.6.2
Next, use the negative value of the to find the second solution.
Step 2.6.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 3
Replace with to show the final answer.
Step 4
Verify if is the inverse of .
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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 range of .
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Step 4.2.1
The range is the set of all valid values. Use the graph to find the range.
Interval Notation:
Step 4.3
Find the domain of .
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Step 4.3.1
Set the radicand in greater than or equal to to find where the expression is defined.
Step 4.3.2
Solve for .
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Step 4.3.2.1
Subtract from both sides of the inequality.
Step 4.3.2.2
Since the expression on each side of the equation has the same denominator, the numerators must be equal.
Step 4.3.2.3
Divide each term in by and simplify.
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Step 4.3.2.3.1
Divide each term in by . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.
Step 4.3.2.3.2
Simplify the left side.
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Step 4.3.2.3.2.1
Dividing two negative values results in a positive value.
Step 4.3.2.3.2.2
Divide by .
Step 4.3.2.3.3
Simplify the right side.
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Step 4.3.2.3.3.1
Divide by .
Step 4.3.2.4
The range of sine is . Since does not fall in this range, there is no solution.
No solution
No solution
Step 4.3.3
The domain is all real numbers.
Step 4.4
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