Trigonometry Examples

Find the Inverse cos(arcsin(7x))
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 cosine of both sides of the equation to extract from inside the cosine.
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
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
Divide each term in by and simplify.
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Step 2.5.1
Divide each term in by .
Step 2.5.2
Simplify the left side.
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Step 2.5.2.1
Cancel the common factor of .
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Step 2.5.2.1.1
Cancel the common factor.
Step 2.5.2.1.2
Divide by .
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
Remove parentheses.
Step 4.2.4
Simplify the numerator.
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Step 4.2.4.1
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 4.2.4.2
Rewrite as .
Step 4.2.4.3
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.2.4.4
Multiply by .
Step 4.2.4.5
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 4.2.4.6
Rewrite as .
Step 4.2.4.7
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.2.4.8
Multiply by .
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
Combine fractions.
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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 .
Step 4.3.4
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.3.5
Simplify.
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Step 4.3.5.1
Write as a fraction with a common denominator.
Step 4.3.5.2
Combine the numerators over the common denominator.
Step 4.3.5.3
Rewrite in a factored form.
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Step 4.3.5.3.1
Factor out of .
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Step 4.3.5.3.1.1
Factor out of .
Step 4.3.5.3.1.2
Factor out of .
Step 4.3.5.3.2
Reduce the expression by cancelling the common factors.
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Step 4.3.5.3.2.1
Reduce the expression by cancelling the common factors.
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Step 4.3.5.3.2.1.1
Cancel the common factor.
Step 4.3.5.3.2.1.2
Rewrite the expression.
Step 4.3.5.3.2.2
Divide by .
Step 4.3.5.4
Cancel the common factor of .
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Step 4.3.5.4.1
Cancel the common factor.
Step 4.3.5.4.2
Divide by .
Step 4.4
Since and , then is the inverse of .