Trigonometry Examples

Find the Inverse 1/2*(arctan( square root of x))^(-1/2)
Step 1
Interchange the variables.
Step 2
Solve for .
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Step 2.1
Simplify .
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Step 2.1.1
Rewrite the expression using the negative exponent rule .
Step 2.1.2
Combine.
Step 2.1.3
Multiply by .
Step 2.2
Rewrite the equation as .
Step 2.3
Use to rewrite as .
Step 2.4
Find the LCD of the terms in the equation.
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Step 2.4.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 2.4.2
The LCM of one and any expression is the expression.
Step 2.5
Multiply each term in by to eliminate the fractions.
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Step 2.5.1
Multiply each term in by .
Step 2.5.2
Simplify the left side.
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Step 2.5.2.1
Rewrite using the commutative property of multiplication.
Step 2.5.2.2
Cancel the common factor of .
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Step 2.5.2.2.1
Cancel the common factor.
Step 2.5.2.2.2
Rewrite the expression.
Step 2.5.2.3
Cancel the common factor of .
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Step 2.5.2.3.1
Cancel the common factor.
Step 2.5.2.3.2
Rewrite the expression.
Step 2.5.3
Simplify the right side.
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Step 2.5.3.1
Rewrite using the commutative property of multiplication.
Step 2.6
Solve the equation.
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Step 2.6.1
Rewrite the equation as .
Step 2.6.2
Divide each term in by and simplify.
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Step 2.6.2.1
Divide each term in by .
Step 2.6.2.2
Simplify the left side.
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Step 2.6.2.2.1
Cancel the common factor.
Step 2.6.2.2.2
Rewrite the expression.
Step 2.6.2.2.3
Cancel the common factor.
Step 2.6.2.2.4
Divide by .
Step 2.6.3
Raise each side of the equation to the power of to eliminate the fractional exponent on the left side.
Step 2.6.4
Simplify the exponent.
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Step 2.6.4.1
Simplify the left side.
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Step 2.6.4.1.1
Simplify .
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Step 2.6.4.1.1.1
Multiply the exponents in .
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Step 2.6.4.1.1.1.1
Apply the power rule and multiply exponents, .
Step 2.6.4.1.1.1.2
Cancel the common factor of .
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Step 2.6.4.1.1.1.2.1
Cancel the common factor.
Step 2.6.4.1.1.1.2.2
Rewrite the expression.
Step 2.6.4.1.1.2
Simplify.
Step 2.6.4.2
Simplify the right side.
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Step 2.6.4.2.1
Simplify .
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Step 2.6.4.2.1.1
Use the power rule to distribute the exponent.
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Step 2.6.4.2.1.1.1
Apply the product rule to .
Step 2.6.4.2.1.1.2
Apply the product rule to .
Step 2.6.4.2.1.2
One to any power is one.
Step 2.6.4.2.1.3
Raise to the power of .
Step 2.6.5
Solve for .
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Step 2.6.5.1
Take the inverse arctangent of both sides of the equation to extract from inside the arctangent.
Step 2.6.5.2
Raise each side of the equation to the power of to eliminate the fractional exponent on the left side.
Step 2.6.5.3
Simplify the left side.
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Step 2.6.5.3.1
Simplify .
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Step 2.6.5.3.1.1
Multiply the exponents in .
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Step 2.6.5.3.1.1.1
Apply the power rule and multiply exponents, .
Step 2.6.5.3.1.1.2
Cancel the common factor of .
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Step 2.6.5.3.1.1.2.1
Cancel the common factor.
Step 2.6.5.3.1.1.2.2
Rewrite the expression.
Step 2.6.5.3.1.2
Simplify.
Step 3
Replace with to show the final answer.
Step 4
Verify if is the inverse of .
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Step 4.1
To verify the inverse, check if and .
Step 4.2
Evaluate .
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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
Simplify the denominator.
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Step 4.2.3.1
Apply the product rule to .
Step 4.2.3.2
Combine exponents.
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Step 4.2.3.2.1
Rewrite as .
Step 4.2.3.2.2
Rewrite as .
Step 4.2.3.2.3
Raise to the power of .
Step 4.2.3.2.4
Apply the power rule and multiply exponents, .
Step 4.2.3.2.5
Multiply by .
Step 4.2.3.2.6
Multiply the exponents in .
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Step 4.2.3.2.6.1
Apply the power rule and multiply exponents, .
Step 4.2.3.2.6.2
Multiply by .
Step 4.2.3.2.7
Use the power rule to combine exponents.
Step 4.2.3.2.8
Subtract from .
Step 4.2.3.3
Anything raised to is .
Step 4.2.3.4
Multiply the exponents in .
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Step 4.2.3.4.1
Apply the power rule and multiply exponents, .
Step 4.2.3.4.2
Cancel the common factor of .
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Step 4.2.3.4.2.1
Move the leading negative in into the numerator.
Step 4.2.3.4.2.2
Cancel the common factor.
Step 4.2.3.4.2.3
Rewrite the expression.
Step 4.2.3.5
Rewrite the expression using the negative exponent rule .
Step 4.2.3.6
Multiply by .
Step 4.2.4
Multiply the numerator by the reciprocal of the denominator.
Step 4.2.5
Multiply by .
Step 4.2.6
The functions tangent and arctangent are inverses.
Step 4.2.7
Rewrite as .
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Step 4.2.7.1
Use to rewrite as .
Step 4.2.7.2
Apply the power rule and multiply exponents, .
Step 4.2.7.3
Combine and .
Step 4.2.7.4
Cancel the common factor of .
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Step 4.2.7.4.1
Cancel the common factor.
Step 4.2.7.4.2
Rewrite the expression.
Step 4.2.7.5
Simplify.
Step 4.3
Evaluate .
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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
Pull terms out from under the radical, assuming positive real numbers.
Step 4.3.4
Rewrite the expression using the negative exponent rule .
Step 4.3.5
Combine.
Step 4.3.6
Multiply by .
Step 4.4
Since and , then is the inverse of .