Trigonometry Examples

Find the Inverse sec(arctan(x/3))
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 secant of both sides of the equation to extract from inside the secant.
Step 2.3
Take the inverse arctangent of both sides of the equation to extract from inside the arctangent.
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
Write the expression using exponents.
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Step 2.4.1.1.1
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 2.4.1.1.2
Rewrite as .
Step 2.4.1.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 2.5
Multiply both sides of the equation by .
Step 2.6
Simplify the left side.
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Step 2.6.1
Cancel the common factor of .
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Step 2.6.1.1
Cancel the common factor.
Step 2.6.1.2
Rewrite the expression.
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 expression.
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Step 4.2.3.1
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 4.2.3.2
Apply the product rule to .
Step 4.2.3.3
Raise to the power of .
Step 4.2.3.4
Write as a fraction with a common denominator.
Step 4.2.4
Combine the numerators over the common denominator.
Step 4.2.5
Rewrite as .
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Step 4.2.5.1
Factor the perfect power out of .
Step 4.2.5.2
Factor the perfect power out of .
Step 4.2.5.3
Rearrange the fraction .
Step 4.2.6
Pull terms out from under the radical.
Step 4.2.7
Combine and .
Step 4.2.8
Write as a fraction with a common denominator.
Step 4.2.9
Combine the numerators over the common denominator.
Step 4.2.10
Simplify the expression.
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Step 4.2.10.1
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 4.2.10.2
Apply the product rule to .
Step 4.2.10.3
Raise to the power of .
Step 4.2.10.4
Write as a fraction with a common denominator.
Step 4.2.11
Combine the numerators over the common denominator.
Step 4.2.12
Rewrite as .
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Step 4.2.12.1
Factor the perfect power out of .
Step 4.2.12.2
Factor the perfect power out of .
Step 4.2.12.3
Rearrange the fraction .
Step 4.2.13
Pull terms out from under the radical.
Step 4.2.14
Combine and .
Step 4.2.15
To write as a fraction with a common denominator, multiply by .
Step 4.2.16
Simplify terms.
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Step 4.2.16.1
Combine and .
Step 4.2.16.2
Combine the numerators over the common denominator.
Step 4.2.16.3
Multiply by .
Step 4.2.16.4
Multiply by .
Step 4.2.16.5
Multiply by .
Step 4.2.17
Expand using the FOIL Method.
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Step 4.2.17.1
Apply the distributive property.
Step 4.2.17.2
Apply the distributive property.
Step 4.2.17.3
Apply the distributive property.
Step 4.2.18
Simplify terms.
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Step 4.2.18.1
Combine the opposite terms in .
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Step 4.2.18.1.1
Reorder the factors in the terms and .
Step 4.2.18.1.2
Add and .
Step 4.2.18.1.3
Add and .
Step 4.2.18.2
Simplify each term.
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Step 4.2.18.2.1
Multiply .
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Step 4.2.18.2.1.1
Raise to the power of .
Step 4.2.18.2.1.2
Raise to the power of .
Step 4.2.18.2.1.3
Use the power rule to combine exponents.
Step 4.2.18.2.1.4
Add and .
Step 4.2.18.2.2
Rewrite as .
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Step 4.2.18.2.2.1
Use to rewrite as .
Step 4.2.18.2.2.2
Apply the power rule and multiply exponents, .
Step 4.2.18.2.2.3
Combine and .
Step 4.2.18.2.2.4
Cancel the common factor of .
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Step 4.2.18.2.2.4.1
Cancel the common factor.
Step 4.2.18.2.2.4.2
Rewrite the expression.
Step 4.2.18.2.2.5
Simplify.
Step 4.2.18.2.3
Multiply by .
Step 4.2.18.3
Simplify by adding terms.
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Step 4.2.18.3.1
Combine the opposite terms in .
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Step 4.2.18.3.1.1
Subtract from .
Step 4.2.18.3.1.2
Add and .
Step 4.2.18.3.2
Rewrite as .
Step 4.2.19
Rewrite as .
Step 4.2.20
Pull terms out from under the radical, assuming positive real numbers.
Step 4.2.21
Cancel the common factor of .
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Step 4.2.21.1
Cancel the common factor.
Step 4.2.21.2
Rewrite the expression.
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
Simplify by cancelling exponent with radical.
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Step 4.3.3.1
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 4.3.3.2
Rewrite as .
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Step 4.3.3.2.1
Use to rewrite as .
Step 4.3.3.2.2
Apply the power rule and multiply exponents, .
Step 4.3.3.2.3
Combine and .
Step 4.3.3.2.4
Cancel the common factor of .
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Step 4.3.3.2.4.1
Cancel the common factor.
Step 4.3.3.2.4.2
Rewrite the expression.
Step 4.3.3.2.5
Simplify.
Step 4.3.4
Expand using the FOIL Method.
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Step 4.3.4.1
Apply the distributive property.
Step 4.3.4.2
Apply the distributive property.
Step 4.3.4.3
Apply the distributive property.
Step 4.3.5
Simplify and combine like terms.
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Step 4.3.5.1
Simplify each term.
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Step 4.3.5.1.1
Multiply by .
Step 4.3.5.1.2
Move to the left of .
Step 4.3.5.1.3
Rewrite as .
Step 4.3.5.1.4
Multiply by .
Step 4.3.5.1.5
Multiply by .
Step 4.3.5.2
Add and .
Step 4.3.5.3
Add and .
Step 4.3.6
Subtract from .
Step 4.3.7
Add and .
Step 4.3.8
Pull terms out from under the radical, assuming positive real numbers.
Step 4.4
Since and , then is the inverse of .