Trigonometry Examples

Find the Inverse tan(arcsin(x))
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 tangent of both sides of the equation to extract from inside the tangent.
Step 2.3
Take the inverse arcsine of both sides of the equation to extract from inside the arcsine.
Step 2.4
Simplify the right side.
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Step 2.4.1
Simplify .
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Step 2.4.1.1
Draw a triangle in the plane with vertices , , and the origin. Then is the angle between the positive x-axis and the ray beginning at the origin and passing through . Therefore, is .
Step 2.4.1.2
Multiply by .
Step 2.4.1.3
Combine and simplify the denominator.
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Step 2.4.1.3.1
Multiply by .
Step 2.4.1.3.2
Raise to the power of .
Step 2.4.1.3.3
Raise to the power of .
Step 2.4.1.3.4
Use the power rule to combine exponents.
Step 2.4.1.3.5
Add and .
Step 2.4.1.3.6
Rewrite as .
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Step 2.4.1.3.6.1
Use to rewrite as .
Step 2.4.1.3.6.2
Apply the power rule and multiply exponents, .
Step 2.4.1.3.6.3
Combine and .
Step 2.4.1.3.6.4
Cancel the common factor of .
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Step 2.4.1.3.6.4.1
Cancel the common factor.
Step 2.4.1.3.6.4.2
Rewrite the expression.
Step 2.4.1.3.6.5
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
Rearrange terms.
Step 4.2.4
Apply pythagorean identity.
Step 4.2.5
Factor out of .
Step 4.2.6
Separate fractions.
Step 4.2.7
Rewrite in terms of sines and cosines.
Step 4.2.8
Rewrite in terms of sines and cosines.
Step 4.2.9
Multiply by the reciprocal of the fraction to divide by .
Step 4.2.10
Reduce the expression by cancelling the common factors.
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Step 4.2.10.1
Write as a fraction with denominator .
Step 4.2.10.2
Cancel the common factor of .
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Step 4.2.10.2.1
Cancel the common factor.
Step 4.2.10.2.2
Rewrite the expression.
Step 4.2.11
Simplify the numerator.
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Step 4.2.11.1
Draw a triangle in the plane with vertices , , and the origin. Then is the angle between the positive x-axis and the ray beginning at the origin and passing through . Therefore, is .
Step 4.2.11.2
Simplify the denominator.
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Step 4.2.11.2.1
Rewrite as .
Step 4.2.11.2.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.2.11.3
Multiply by .
Step 4.2.11.4
Combine and simplify the denominator.
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Step 4.2.11.4.1
Multiply by .
Step 4.2.11.4.2
Raise to the power of .
Step 4.2.11.4.3
Raise to the power of .
Step 4.2.11.4.4
Use the power rule to combine exponents.
Step 4.2.11.4.5
Add and .
Step 4.2.11.4.6
Rewrite as .
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Step 4.2.11.4.6.1
Use to rewrite as .
Step 4.2.11.4.6.2
Apply the power rule and multiply exponents, .
Step 4.2.11.4.6.3
Combine and .
Step 4.2.11.4.6.4
Cancel the common factor of .
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Step 4.2.11.4.6.4.1
Cancel the common factor.
Step 4.2.11.4.6.4.2
Rewrite the expression.
Step 4.2.11.4.6.5
Simplify.
Step 4.2.11.5
Use the power rule to distribute the exponent.
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Step 4.2.11.5.1
Apply the product rule to .
Step 4.2.11.5.2
Apply the product rule to .
Step 4.2.11.5.3
Apply the product rule to .
Step 4.2.11.6
Simplify the numerator.
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Step 4.2.11.6.1
Rewrite as .
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Step 4.2.11.6.1.1
Use to rewrite as .
Step 4.2.11.6.1.2
Apply the power rule and multiply exponents, .
Step 4.2.11.6.1.3
Combine and .
Step 4.2.11.6.1.4
Cancel the common factor of .
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Step 4.2.11.6.1.4.1
Cancel the common factor.
Step 4.2.11.6.1.4.2
Rewrite the expression.
Step 4.2.11.6.1.5
Simplify.
Step 4.2.11.6.2
Expand using the FOIL Method.
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Step 4.2.11.6.2.1
Apply the distributive property.
Step 4.2.11.6.2.2
Apply the distributive property.
Step 4.2.11.6.2.3
Apply the distributive property.
Step 4.2.11.6.3
Simplify and combine like terms.
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Step 4.2.11.6.3.1
Simplify each term.
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Step 4.2.11.6.3.1.1
Multiply by .
Step 4.2.11.6.3.1.2
Multiply by .
Step 4.2.11.6.3.1.3
Multiply by .
Step 4.2.11.6.3.1.4
Rewrite using the commutative property of multiplication.
Step 4.2.11.6.3.1.5
Multiply by by adding the exponents.
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Step 4.2.11.6.3.1.5.1
Move .
Step 4.2.11.6.3.1.5.2
Multiply by .
Step 4.2.11.6.3.2
Add and .
Step 4.2.11.6.3.3
Add and .
Step 4.2.11.6.4
Rewrite as .
Step 4.2.11.6.5
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.2.11.7
Cancel the common factor of and .
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Step 4.2.11.7.1
Factor out of .
Step 4.2.11.7.2
Cancel the common factors.
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Step 4.2.11.7.2.1
Factor out of .
Step 4.2.11.7.2.2
Cancel the common factor.
Step 4.2.11.7.2.3
Rewrite the expression.
Step 4.2.11.8
Cancel the common factor of and .
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Step 4.2.11.8.1
Factor out of .
Step 4.2.11.8.2
Cancel the common factors.
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Step 4.2.11.8.2.1
Factor out of .
Step 4.2.11.8.2.2
Cancel the common factor.
Step 4.2.11.8.2.3
Rewrite the expression.
Step 4.2.11.9
Write as a fraction with a common denominator.
Step 4.2.11.10
Combine the numerators over the common denominator.
Step 4.2.11.11
Simplify the numerator.
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Step 4.2.11.11.1
Expand using the FOIL Method.
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Step 4.2.11.11.1.1
Apply the distributive property.
Step 4.2.11.11.1.2
Apply the distributive property.
Step 4.2.11.11.1.3
Apply the distributive property.
Step 4.2.11.11.2
Simplify and combine like terms.
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Step 4.2.11.11.2.1
Simplify each term.
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Step 4.2.11.11.2.1.1
Multiply by .
Step 4.2.11.11.2.1.2
Multiply by .
Step 4.2.11.11.2.1.3
Multiply by .
Step 4.2.11.11.2.1.4
Rewrite using the commutative property of multiplication.
Step 4.2.11.11.2.1.5
Multiply by by adding the exponents.
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Step 4.2.11.11.2.1.5.1
Move .
Step 4.2.11.11.2.1.5.2
Multiply by .
Step 4.2.11.11.2.2
Add and .
Step 4.2.11.11.2.3
Add and .
Step 4.2.11.11.3
Add and .
Step 4.2.11.11.4
Add and .
Step 4.2.11.12
Rewrite as .
Step 4.2.11.13
Any root of is .
Step 4.2.11.14
Multiply by .
Step 4.2.11.15
Combine and simplify the denominator.
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Step 4.2.11.15.1
Multiply by .
Step 4.2.11.15.2
Raise to the power of .
Step 4.2.11.15.3
Raise to the power of .
Step 4.2.11.15.4
Use the power rule to combine exponents.
Step 4.2.11.15.5
Add and .
Step 4.2.11.15.6
Rewrite as .
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Step 4.2.11.15.6.1
Use to rewrite as .
Step 4.2.11.15.6.2
Apply the power rule and multiply exponents, .
Step 4.2.11.15.6.3
Combine and .
Step 4.2.11.15.6.4
Cancel the common factor of .
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Step 4.2.11.15.6.4.1
Cancel the common factor.
Step 4.2.11.15.6.4.2
Rewrite the expression.
Step 4.2.11.15.6.5
Simplify.
Step 4.2.12
Simplify the denominator.
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Step 4.2.12.1
Draw a triangle in the plane with vertices , , and the origin. Then is the angle between the positive x-axis and the ray beginning at the origin and passing through . Therefore, is .
Step 4.2.12.2
Simplify the denominator.
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Step 4.2.12.2.1
Rewrite as .
Step 4.2.12.2.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.2.12.3
Multiply by .
Step 4.2.12.4
Combine and simplify the denominator.
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Step 4.2.12.4.1
Multiply by .
Step 4.2.12.4.2
Raise to the power of .
Step 4.2.12.4.3
Raise to the power of .
Step 4.2.12.4.4
Use the power rule to combine exponents.
Step 4.2.12.4.5
Add and .
Step 4.2.12.4.6
Rewrite as .
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Step 4.2.12.4.6.1
Use to rewrite as .
Step 4.2.12.4.6.2
Apply the power rule and multiply exponents, .
Step 4.2.12.4.6.3
Combine and .
Step 4.2.12.4.6.4
Cancel the common factor of .
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Step 4.2.12.4.6.4.1
Cancel the common factor.
Step 4.2.12.4.6.4.2
Rewrite the expression.
Step 4.2.12.4.6.5
Simplify.
Step 4.2.13
Reduce the expression by cancelling the common factors.
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Step 4.2.13.1
Cancel the common factor of .
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Step 4.2.13.1.1
Cancel the common factor.
Step 4.2.13.1.2
Rewrite the expression.
Step 4.2.13.2
Multiply by .
Step 4.2.14
The functions sine and arcsine are inverses.
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
Draw a triangle in the plane with vertices , , and the origin. Then is the angle between the positive x-axis and the ray beginning at the origin and passing through . Therefore, is .
Step 4.3.4
Multiply the numerator by the reciprocal of the denominator.
Step 4.3.5
Simplify the denominator.
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Step 4.3.5.1
Rewrite as .
Step 4.3.5.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.3.5.3
Simplify.
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Step 4.3.5.3.1
Write as a fraction with a common denominator.
Step 4.3.5.3.2
Combine the numerators over the common denominator.
Step 4.3.5.3.3
Write as a fraction with a common denominator.
Step 4.3.5.3.4
Combine the numerators over the common denominator.
Step 4.3.5.4
Multiply by .
Step 4.3.5.5
Simplify the denominator.
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Step 4.3.5.5.1
Raise to the power of .
Step 4.3.5.5.2
Raise to the power of .
Step 4.3.5.5.3
Use the power rule to combine exponents.
Step 4.3.5.5.4
Add and .
Step 4.3.5.6
Rewrite as .
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Step 4.3.5.6.1
Factor the perfect power out of .
Step 4.3.5.6.2
Factor the perfect power out of .
Step 4.3.5.6.3
Rearrange the fraction .
Step 4.3.5.7
Pull terms out from under the radical.
Step 4.3.5.8
Combine and .
Step 4.3.6
Multiply the numerator by the reciprocal of the denominator.
Step 4.3.7
Multiply by .
Step 4.3.8
Cancel the common factor of .
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Step 4.3.8.1
Cancel the common factor.
Step 4.3.8.2
Rewrite the expression.
Step 4.3.9
Combine and .
Step 4.3.10
Combine and .
Step 4.3.11
Multiply by .
Step 4.3.12
Combine and simplify the denominator.
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Step 4.3.12.1
Multiply by .
Step 4.3.12.2
Raise to the power of .
Step 4.3.12.3
Raise to the power of .
Step 4.3.12.4
Use the power rule to combine exponents.
Step 4.3.12.5
Add and .
Step 4.3.12.6
Rewrite as .
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Step 4.3.12.6.1
Use to rewrite as .
Step 4.3.12.6.2
Apply the power rule and multiply exponents, .
Step 4.3.12.6.3
Combine and .
Step 4.3.12.6.4
Cancel the common factor of .
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Step 4.3.12.6.4.1
Cancel the common factor.
Step 4.3.12.6.4.2
Rewrite the expression.
Step 4.3.12.6.5
Simplify.
Step 4.3.13
Combine using the product rule for radicals.
Step 4.3.14
Multiply by .
Step 4.3.15
Multiply by .
Step 4.3.16
Reorder.
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Step 4.3.16.1
Move .
Step 4.3.16.2
Expand the denominator using the FOIL method.
Step 4.3.16.3
Simplify.
Step 4.3.17
Cancel the common factor of .
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Step 4.3.17.1
Cancel the common factor.
Step 4.3.17.2
Rewrite the expression.
Step 4.4
Since and , then is the inverse of .