Trigonometry Examples

Find the Inverse f(x)=7arcsin(x^2)
Step 1
Write as an equation.
Step 2
Interchange the variables.
Step 3
Solve for .
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Step 3.1
Rewrite the equation as .
Step 3.2
Divide each term in by and simplify.
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Step 3.2.1
Divide each term in by .
Step 3.2.2
Simplify the left side.
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Step 3.2.2.1
Cancel the common factor of .
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Step 3.2.2.1.1
Cancel the common factor.
Step 3.2.2.1.2
Divide by .
Step 3.3
Take the inverse arcsine of both sides of the equation to extract from inside the arcsine.
Step 3.4
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 3.5
The complete solution is the result of both the positive and negative portions of the solution.
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Step 3.5.1
First, use the positive value of the to find the first solution.
Step 3.5.2
Next, use the negative value of the to find the second solution.
Step 3.5.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 4
Replace with to show the final answer.
Step 5
Verify if is the inverse of .
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Step 5.1
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 5.2
Find the range of .
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Step 5.2.1
The range is the set of all valid values. Use the graph to find the range.
Interval Notation:
Step 5.3
Find the domain of the inverse.
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Step 5.3.1
Find the domain of .
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Step 5.3.1.1
Set the radicand in greater than or equal to to find where the expression is defined.
Step 5.3.1.2
Solve for .
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Step 5.3.1.2.1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 5.3.1.2.2
Simplify the right side.
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Step 5.3.1.2.2.1
The exact value of is .
Step 5.3.1.2.3
Multiply both sides by .
Step 5.3.1.2.4
Simplify.
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Step 5.3.1.2.4.1
Simplify the left side.
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Step 5.3.1.2.4.1.1
Cancel the common factor of .
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Step 5.3.1.2.4.1.1.1
Cancel the common factor.
Step 5.3.1.2.4.1.1.2
Rewrite the expression.
Step 5.3.1.2.4.2
Simplify the right side.
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Step 5.3.1.2.4.2.1
Multiply by .
Step 5.3.1.2.5
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 5.3.1.2.6
Solve for .
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Step 5.3.1.2.6.1
Multiply both sides of the equation by .
Step 5.3.1.2.6.2
Simplify both sides of the equation.
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Step 5.3.1.2.6.2.1
Simplify the left side.
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Step 5.3.1.2.6.2.1.1
Cancel the common factor of .
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Step 5.3.1.2.6.2.1.1.1
Cancel the common factor.
Step 5.3.1.2.6.2.1.1.2
Rewrite the expression.
Step 5.3.1.2.6.2.2
Simplify the right side.
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Step 5.3.1.2.6.2.2.1
Subtract from .
Step 5.3.1.2.7
Find the period of .
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Step 5.3.1.2.7.1
The period of the function can be calculated using .
Step 5.3.1.2.7.2
Replace with in the formula for period.
Step 5.3.1.2.7.3
is approximately which is positive so remove the absolute value
Step 5.3.1.2.7.4
Multiply the numerator by the reciprocal of the denominator.
Step 5.3.1.2.7.5
Multiply by .
Step 5.3.1.2.8
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 5.3.1.2.9
Consolidate the answers.
, for any integer
Step 5.3.1.2.10
Use each root to create test intervals.
Step 5.3.1.2.11
Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.
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Step 5.3.1.2.11.1
Test a value on the interval to see if it makes the inequality true.
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Step 5.3.1.2.11.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 5.3.1.2.11.1.2
Replace with in the original inequality.
Step 5.3.1.2.11.1.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 5.3.1.2.11.2
Test a value on the interval to see if it makes the inequality true.
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Step 5.3.1.2.11.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 5.3.1.2.11.2.2
Replace with in the original inequality.
Step 5.3.1.2.11.2.3
The left side is less than the right side , which means that the given statement is false.
False
False
Step 5.3.1.2.11.3
Compare the intervals to determine which ones satisfy the original inequality.
True
False
True
False
Step 5.3.1.2.12
The solution consists of all of the true intervals.
, for any integer
, for any integer
Step 5.3.1.3
The domain is all values of that make the expression defined.
, for any integer
, for any integer
Step 5.3.2

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Step 5.3.2.1
The union consists of all of the elements that are contained in each interval.
No solution
No solution
No solution
Step 5.4
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 6