Trigonometry Examples

Find the Inverse square root of 5x^2-10x+52
Step 1
Interchange the variables.
Step 2
Solve for .
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Step 2.1
Rewrite the equation as .
Step 2.2
Move all terms not containing to the right side of the equation.
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Step 2.2.1
Add to both sides of the equation.
Step 2.2.2
Subtract from both sides of the equation.
Step 2.3
To remove the radical on the left side of the equation, square both sides of the equation.
Step 2.4
Simplify each side of the equation.
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Step 2.4.1
Use to rewrite as .
Step 2.4.2
Simplify the left side.
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Step 2.4.2.1
Simplify .
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Step 2.4.2.1.1
Multiply the exponents in .
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Step 2.4.2.1.1.1
Apply the power rule and multiply exponents, .
Step 2.4.2.1.1.2
Cancel the common factor of .
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Step 2.4.2.1.1.2.1
Cancel the common factor.
Step 2.4.2.1.1.2.2
Rewrite the expression.
Step 2.4.2.1.2
Simplify.
Step 2.4.3
Simplify the right side.
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Step 2.4.3.1
Simplify .
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Step 2.4.3.1.1
Rewrite as .
Step 2.4.3.1.2
Expand by multiplying each term in the first expression by each term in the second expression.
Step 2.4.3.1.3
Simplify each term.
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Step 2.4.3.1.3.1
Multiply by .
Step 2.4.3.1.3.2
Rewrite using the commutative property of multiplication.
Step 2.4.3.1.3.3
Move to the left of .
Step 2.4.3.1.3.4
Rewrite using the commutative property of multiplication.
Step 2.4.3.1.3.5
Multiply by by adding the exponents.
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Step 2.4.3.1.3.5.1
Move .
Step 2.4.3.1.3.5.2
Multiply by .
Step 2.4.3.1.3.6
Multiply by .
Step 2.4.3.1.3.7
Multiply by .
Step 2.4.3.1.3.8
Multiply by .
Step 2.4.3.1.3.9
Multiply by .
Step 2.4.3.1.4
Add and .
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Step 2.4.3.1.4.1
Move .
Step 2.4.3.1.4.2
Add and .
Step 2.4.3.1.5
Subtract from .
Step 2.4.3.1.6
Subtract from .
Step 2.5
Solve for .
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Step 2.5.1
Since is on the right side of the equation, switch the sides so it is on the left side of the equation.
Step 2.5.2
Move all terms containing to the left side of the equation.
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Step 2.5.2.1
Subtract from both sides of the equation.
Step 2.5.2.2
Subtract from .
Step 2.5.3
Use the quadratic formula to find the solutions.
Step 2.5.4
Substitute the values , , and into the quadratic formula and solve for .
Step 2.5.5
Simplify.
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Step 2.5.5.1
Simplify the numerator.
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Step 2.5.5.1.1
Apply the distributive property.
Step 2.5.5.1.2
Multiply by .
Step 2.5.5.1.3
Multiply by .
Step 2.5.5.1.4
Add parentheses.
Step 2.5.5.1.5
Let . Substitute for all occurrences of .
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Step 2.5.5.1.5.1
Rewrite as .
Step 2.5.5.1.5.2
Expand using the FOIL Method.
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Step 2.5.5.1.5.2.1
Apply the distributive property.
Step 2.5.5.1.5.2.2
Apply the distributive property.
Step 2.5.5.1.5.2.3
Apply the distributive property.
Step 2.5.5.1.5.3
Simplify and combine like terms.
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Step 2.5.5.1.5.3.1
Simplify each term.
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Step 2.5.5.1.5.3.1.1
Rewrite using the commutative property of multiplication.
Step 2.5.5.1.5.3.1.2
Multiply by by adding the exponents.
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Step 2.5.5.1.5.3.1.2.1
Move .
Step 2.5.5.1.5.3.1.2.2
Multiply by .
Step 2.5.5.1.5.3.1.3
Multiply by .
Step 2.5.5.1.5.3.1.4
Multiply by .
Step 2.5.5.1.5.3.1.5
Multiply by .
Step 2.5.5.1.5.3.1.6
Multiply by .
Step 2.5.5.1.5.3.2
Subtract from .
Step 2.5.5.1.6
Factor out of .
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Step 2.5.5.1.6.1
Factor out of .
Step 2.5.5.1.6.2
Factor out of .
Step 2.5.5.1.6.3
Factor out of .
Step 2.5.5.1.6.4
Factor out of .
Step 2.5.5.1.6.5
Factor out of .
Step 2.5.5.1.6.6
Factor out of .
Step 2.5.5.1.6.7
Factor out of .
Step 2.5.5.1.7
Replace all occurrences of with .
Step 2.5.5.1.8
Simplify.
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Step 2.5.5.1.8.1
Simplify each term.
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Step 2.5.5.1.8.1.1
Apply the distributive property.
Step 2.5.5.1.8.1.2
Simplify.
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Step 2.5.5.1.8.1.2.1
Multiply by .
Step 2.5.5.1.8.1.2.2
Multiply by .
Step 2.5.5.1.8.1.3
Apply the distributive property.
Step 2.5.5.1.8.1.4
Simplify.
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Step 2.5.5.1.8.1.4.1
Multiply by .
Step 2.5.5.1.8.1.4.2
Multiply by .
Step 2.5.5.1.8.1.4.3
Multiply by .
Step 2.5.5.1.8.2
Subtract from .
Step 2.5.5.1.8.3
Add and .
Step 2.5.5.1.8.4
Subtract from .
Step 2.5.5.1.9
Factor out of .
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Step 2.5.5.1.9.1
Factor out of .
Step 2.5.5.1.9.2
Factor out of .
Step 2.5.5.1.9.3
Factor out of .
Step 2.5.5.1.9.4
Factor out of .
Step 2.5.5.1.9.5
Factor out of .
Step 2.5.5.1.10
Factor using the perfect square rule.
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Step 2.5.5.1.10.1
Rewrite as .
Step 2.5.5.1.10.2
Check that the middle term is two times the product of the numbers being squared in the first term and third term.
Step 2.5.5.1.10.3
Rewrite the polynomial.
Step 2.5.5.1.10.4
Factor using the perfect square trinomial rule , where and .
Step 2.5.5.1.11
Multiply by .
Step 2.5.5.1.12
Rewrite as .
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Step 2.5.5.1.12.1
Factor out of .
Step 2.5.5.1.12.2
Rewrite as .
Step 2.5.5.1.12.3
Move .
Step 2.5.5.1.12.4
Rewrite as .
Step 2.5.5.1.13
Pull terms out from under the radical.
Step 2.5.5.1.14
Apply the distributive property.
Step 2.5.5.1.15
Multiply by .
Step 2.5.5.1.16
Apply the distributive property.
Step 2.5.5.2
Multiply by .
Step 2.5.6
Simplify the expression to solve for the portion of the .
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Step 2.5.6.1
Simplify the numerator.
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Step 2.5.6.1.1
Apply the distributive property.
Step 2.5.6.1.2
Multiply by .
Step 2.5.6.1.3
Multiply by .
Step 2.5.6.1.4
Add parentheses.
Step 2.5.6.1.5
Let . Substitute for all occurrences of .
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Step 2.5.6.1.5.1
Rewrite as .
Step 2.5.6.1.5.2
Expand using the FOIL Method.
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Step 2.5.6.1.5.2.1
Apply the distributive property.
Step 2.5.6.1.5.2.2
Apply the distributive property.
Step 2.5.6.1.5.2.3
Apply the distributive property.
Step 2.5.6.1.5.3
Simplify and combine like terms.
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Step 2.5.6.1.5.3.1
Simplify each term.
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Step 2.5.6.1.5.3.1.1
Rewrite using the commutative property of multiplication.
Step 2.5.6.1.5.3.1.2
Multiply by by adding the exponents.
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Step 2.5.6.1.5.3.1.2.1
Move .
Step 2.5.6.1.5.3.1.2.2
Multiply by .
Step 2.5.6.1.5.3.1.3
Multiply by .
Step 2.5.6.1.5.3.1.4
Multiply by .
Step 2.5.6.1.5.3.1.5
Multiply by .
Step 2.5.6.1.5.3.1.6
Multiply by .
Step 2.5.6.1.5.3.2
Subtract from .
Step 2.5.6.1.6
Factor out of .
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Step 2.5.6.1.6.1
Factor out of .
Step 2.5.6.1.6.2
Factor out of .
Step 2.5.6.1.6.3
Factor out of .
Step 2.5.6.1.6.4
Factor out of .
Step 2.5.6.1.6.5
Factor out of .
Step 2.5.6.1.6.6
Factor out of .
Step 2.5.6.1.6.7
Factor out of .
Step 2.5.6.1.7
Replace all occurrences of with .
Step 2.5.6.1.8
Simplify.
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Step 2.5.6.1.8.1
Simplify each term.
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Step 2.5.6.1.8.1.1
Apply the distributive property.
Step 2.5.6.1.8.1.2
Simplify.
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Step 2.5.6.1.8.1.2.1
Multiply by .
Step 2.5.6.1.8.1.2.2
Multiply by .
Step 2.5.6.1.8.1.3
Apply the distributive property.
Step 2.5.6.1.8.1.4
Simplify.
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Step 2.5.6.1.8.1.4.1
Multiply by .
Step 2.5.6.1.8.1.4.2
Multiply by .
Step 2.5.6.1.8.1.4.3
Multiply by .
Step 2.5.6.1.8.2
Subtract from .
Step 2.5.6.1.8.3
Add and .
Step 2.5.6.1.8.4
Subtract from .
Step 2.5.6.1.9
Factor out of .
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Step 2.5.6.1.9.1
Factor out of .
Step 2.5.6.1.9.2
Factor out of .
Step 2.5.6.1.9.3
Factor out of .
Step 2.5.6.1.9.4
Factor out of .
Step 2.5.6.1.9.5
Factor out of .
Step 2.5.6.1.10
Factor using the perfect square rule.
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Step 2.5.6.1.10.1
Rewrite as .
Step 2.5.6.1.10.2
Check that the middle term is two times the product of the numbers being squared in the first term and third term.
Step 2.5.6.1.10.3
Rewrite the polynomial.
Step 2.5.6.1.10.4
Factor using the perfect square trinomial rule , where and .
Step 2.5.6.1.11
Multiply by .
Step 2.5.6.1.12
Rewrite as .
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Step 2.5.6.1.12.1
Factor out of .
Step 2.5.6.1.12.2
Rewrite as .
Step 2.5.6.1.12.3
Move .
Step 2.5.6.1.12.4
Rewrite as .
Step 2.5.6.1.13
Pull terms out from under the radical.
Step 2.5.6.1.14
Apply the distributive property.
Step 2.5.6.1.15
Multiply by .
Step 2.5.6.1.16
Apply the distributive property.
Step 2.5.6.2
Multiply by .
Step 2.5.6.3
Change the to .
Step 2.5.6.4
Cancel the common factor of and .
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Step 2.5.6.4.1
Factor out of .
Step 2.5.6.4.2
Factor out of .
Step 2.5.6.4.3
Factor out of .
Step 2.5.6.4.4
Factor out of .
Step 2.5.6.4.5
Factor out of .
Step 2.5.6.4.6
Factor out of .
Step 2.5.6.4.7
Factor out of .
Step 2.5.6.4.8
Cancel the common factors.
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Step 2.5.6.4.8.1
Factor out of .
Step 2.5.6.4.8.2
Cancel the common factor.
Step 2.5.6.4.8.3
Rewrite the expression.
Step 2.5.6.5
Reorder terms.
Step 2.5.7
Simplify the expression to solve for the portion of the .
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Step 2.5.7.1
Simplify the numerator.
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Step 2.5.7.1.1
Apply the distributive property.
Step 2.5.7.1.2
Multiply by .
Step 2.5.7.1.3
Multiply by .
Step 2.5.7.1.4
Add parentheses.
Step 2.5.7.1.5
Let . Substitute for all occurrences of .
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Step 2.5.7.1.5.1
Rewrite as .
Step 2.5.7.1.5.2
Expand using the FOIL Method.
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Step 2.5.7.1.5.2.1
Apply the distributive property.
Step 2.5.7.1.5.2.2
Apply the distributive property.
Step 2.5.7.1.5.2.3
Apply the distributive property.
Step 2.5.7.1.5.3
Simplify and combine like terms.
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Step 2.5.7.1.5.3.1
Simplify each term.
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Step 2.5.7.1.5.3.1.1
Rewrite using the commutative property of multiplication.
Step 2.5.7.1.5.3.1.2
Multiply by by adding the exponents.
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Step 2.5.7.1.5.3.1.2.1
Move .
Step 2.5.7.1.5.3.1.2.2
Multiply by .
Step 2.5.7.1.5.3.1.3
Multiply by .
Step 2.5.7.1.5.3.1.4
Multiply by .
Step 2.5.7.1.5.3.1.5
Multiply by .
Step 2.5.7.1.5.3.1.6
Multiply by .
Step 2.5.7.1.5.3.2
Subtract from .
Step 2.5.7.1.6
Factor out of .
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Step 2.5.7.1.6.1
Factor out of .
Step 2.5.7.1.6.2
Factor out of .
Step 2.5.7.1.6.3
Factor out of .
Step 2.5.7.1.6.4
Factor out of .
Step 2.5.7.1.6.5
Factor out of .
Step 2.5.7.1.6.6
Factor out of .
Step 2.5.7.1.6.7
Factor out of .
Step 2.5.7.1.7
Replace all occurrences of with .
Step 2.5.7.1.8
Simplify.
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Step 2.5.7.1.8.1
Simplify each term.
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Step 2.5.7.1.8.1.1
Apply the distributive property.
Step 2.5.7.1.8.1.2
Simplify.
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Step 2.5.7.1.8.1.2.1
Multiply by .
Step 2.5.7.1.8.1.2.2
Multiply by .
Step 2.5.7.1.8.1.3
Apply the distributive property.
Step 2.5.7.1.8.1.4
Simplify.
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Step 2.5.7.1.8.1.4.1
Multiply by .
Step 2.5.7.1.8.1.4.2
Multiply by .
Step 2.5.7.1.8.1.4.3
Multiply by .
Step 2.5.7.1.8.2
Subtract from .
Step 2.5.7.1.8.3
Add and .
Step 2.5.7.1.8.4
Subtract from .
Step 2.5.7.1.9
Factor out of .
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Step 2.5.7.1.9.1
Factor out of .
Step 2.5.7.1.9.2
Factor out of .
Step 2.5.7.1.9.3
Factor out of .
Step 2.5.7.1.9.4
Factor out of .
Step 2.5.7.1.9.5
Factor out of .
Step 2.5.7.1.10
Factor using the perfect square rule.
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Step 2.5.7.1.10.1
Rewrite as .
Step 2.5.7.1.10.2
Check that the middle term is two times the product of the numbers being squared in the first term and third term.
Step 2.5.7.1.10.3
Rewrite the polynomial.
Step 2.5.7.1.10.4
Factor using the perfect square trinomial rule , where and .
Step 2.5.7.1.11
Multiply by .
Step 2.5.7.1.12
Rewrite as .
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Step 2.5.7.1.12.1
Factor out of .
Step 2.5.7.1.12.2
Rewrite as .
Step 2.5.7.1.12.3
Move .
Step 2.5.7.1.12.4
Rewrite as .
Step 2.5.7.1.13
Pull terms out from under the radical.
Step 2.5.7.1.14
Apply the distributive property.
Step 2.5.7.1.15
Multiply by .
Step 2.5.7.1.16
Apply the distributive property.
Step 2.5.7.2
Multiply by .
Step 2.5.7.3
Change the to .
Step 2.5.7.4
Cancel the common factor of and .
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Step 2.5.7.4.1
Factor out of .
Step 2.5.7.4.2
Rewrite as .
Step 2.5.7.4.3
Factor out of .
Step 2.5.7.4.4
Factor out of .
Step 2.5.7.4.5
Rewrite as .
Step 2.5.7.4.6
Factor out of .
Step 2.5.7.4.7
Cancel the common factors.
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Step 2.5.7.4.7.1
Factor out of .
Step 2.5.7.4.7.2
Cancel the common factor.
Step 2.5.7.4.7.3
Rewrite the expression.
Step 2.5.7.5
Reorder terms.
Step 2.5.7.6
Move the negative in front of the fraction.
Step 2.5.8
The final answer is the combination of both solutions.
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 .
Step 4.4
Since the domain of is the range of and the range of is the domain of , then is the inverse of .
Step 5