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Trigonometry Examples
Step 1
Write as an equation.
Step 2
Interchange the variables.
Step 3
Step 3.1
Rewrite the equation as .
Step 3.2
Subtract from both sides of the equation.
Step 3.3
Divide each term in by and simplify.
Step 3.3.1
Divide each term in by .
Step 3.3.2
Simplify the left side.
Step 3.3.2.1
Cancel the common factor of .
Step 3.3.2.1.1
Cancel the common factor.
Step 3.3.2.1.2
Divide by .
Step 3.3.3
Simplify the right side.
Step 3.3.3.1
Simplify each term.
Step 3.3.3.1.1
Cancel the common factor of and .
Step 3.3.3.1.1.1
Factor out of .
Step 3.3.3.1.1.2
Cancel the common factors.
Step 3.3.3.1.1.2.1
Factor out of .
Step 3.3.3.1.1.2.2
Cancel the common factor.
Step 3.3.3.1.1.2.3
Rewrite the expression.
Step 3.3.3.1.2
Move the negative in front of the fraction.
Step 3.4
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 3.5
Simplify the left side.
Step 3.5.1
Combine and .
Step 3.6
Add to both sides of the equation.
Step 3.7
Multiply both sides of the equation by .
Step 3.8
Simplify both sides of the equation.
Step 3.8.1
Simplify the left side.
Step 3.8.1.1
Simplify .
Step 3.8.1.1.1
Cancel the common factor of .
Step 3.8.1.1.1.1
Cancel the common factor.
Step 3.8.1.1.1.2
Rewrite the expression.
Step 3.8.1.1.2
Cancel the common factor of .
Step 3.8.1.1.2.1
Factor out of .
Step 3.8.1.1.2.2
Cancel the common factor.
Step 3.8.1.1.2.3
Rewrite the expression.
Step 3.8.2
Simplify the right side.
Step 3.8.2.1
Simplify .
Step 3.8.2.1.1
Apply the distributive property.
Step 3.8.2.1.2
Combine and .
Step 3.8.2.1.3
Cancel the common factor of .
Step 3.8.2.1.3.1
Factor out of .
Step 3.8.2.1.3.2
Cancel the common factor.
Step 3.8.2.1.3.3
Rewrite the expression.
Step 3.8.2.1.4
Cancel the common factor of .
Step 3.8.2.1.4.1
Factor out of .
Step 3.8.2.1.4.2
Cancel the common factor.
Step 3.8.2.1.4.3
Rewrite the expression.
Step 3.9
Add to both sides of the equation.
Step 3.10
Multiply both sides of the equation by .
Step 3.11
Simplify both sides of the equation.
Step 3.11.1
Simplify the left side.
Step 3.11.1.1
Simplify .
Step 3.11.1.1.1
Cancel the common factor of .
Step 3.11.1.1.1.1
Cancel the common factor.
Step 3.11.1.1.1.2
Rewrite the expression.
Step 3.11.1.1.2
Cancel the common factor of .
Step 3.11.1.1.2.1
Factor out of .
Step 3.11.1.1.2.2
Cancel the common factor.
Step 3.11.1.1.2.3
Rewrite the expression.
Step 3.11.2
Simplify the right side.
Step 3.11.2.1
Simplify .
Step 3.11.2.1.1
To write as a fraction with a common denominator, multiply by .
Step 3.11.2.1.2
Simplify terms.
Step 3.11.2.1.2.1
Combine and .
Step 3.11.2.1.2.2
Combine the numerators over the common denominator.
Step 3.11.2.1.2.3
Cancel the common factor of .
Step 3.11.2.1.2.3.1
Factor out of .
Step 3.11.2.1.2.3.2
Cancel the common factor.
Step 3.11.2.1.2.3.3
Rewrite the expression.
Step 3.11.2.1.3
Move to the left of .
Step 3.11.2.1.4
Apply the distributive property.
Step 3.11.2.1.5
Multiply .
Step 3.11.2.1.5.1
Combine and .
Step 3.11.2.1.5.2
Multiply by .
Step 3.11.2.1.5.3
Combine and .
Step 3.11.2.1.6
Cancel the common factor of .
Step 3.11.2.1.6.1
Factor out of .
Step 3.11.2.1.6.2
Cancel the common factor.
Step 3.11.2.1.6.3
Rewrite the expression.
Step 3.11.2.1.7
Simplify the numerator.
Step 3.11.2.1.7.1
To write as a fraction with a common denominator, multiply by .
Step 3.11.2.1.7.2
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 3.11.2.1.7.2.1
Multiply by .
Step 3.11.2.1.7.2.2
Multiply by .
Step 3.11.2.1.7.3
Combine the numerators over the common denominator.
Step 3.11.2.1.7.4
Multiply by .
Step 3.11.2.1.8
To write as a fraction with a common denominator, multiply by .
Step 3.11.2.1.9
Multiply by .
Step 3.11.2.1.10
Combine the numerators over the common denominator.
Step 3.11.2.1.11
Factor out of .
Step 3.11.2.1.11.1
Factor out of .
Step 3.11.2.1.11.2
Factor out of .
Step 3.11.2.1.11.3
Factor out of .
Step 4
Replace with to show the final answer.
Step 5
Step 5.1
To verify the inverse, check if and .
Step 5.2
Evaluate .
Step 5.2.1
Set up the composite result function.
Step 5.2.2
Evaluate by substituting in the value of into .
Step 5.2.3
Cancel the common factor of and .
Step 5.2.3.1
Factor out of .
Step 5.2.3.2
Factor out of .
Step 5.2.3.3
Factor out of .
Step 5.2.3.4
Factor out of .
Step 5.2.3.5
Factor out of .
Step 5.2.3.6
Cancel the common factors.
Step 5.2.3.6.1
Factor out of .
Step 5.2.3.6.2
Cancel the common factor.
Step 5.2.3.6.3
Rewrite the expression.
Step 5.2.4
Simplify the numerator.
Step 5.2.4.1
Combine and .
Step 5.2.4.2
Subtract from .
Step 5.2.4.3
Add and .
Step 5.2.5
Cancel the common factor of .
Step 5.2.5.1
Cancel the common factor.
Step 5.2.5.2
Divide by .
Step 5.3
Evaluate .
Step 5.3.1
Set up the composite result function.
Step 5.3.2
Evaluate by substituting in the value of into .
Step 5.3.3
Simplify each term.
Step 5.3.3.1
Simplify each term.
Step 5.3.3.1.1
Cancel the common factor of .
Step 5.3.3.1.1.1
Cancel the common factor.
Step 5.3.3.1.1.2
Rewrite the expression.
Step 5.3.3.1.2
Cancel the common factor of .
Step 5.3.3.1.2.1
Factor out of .
Step 5.3.3.1.2.2
Cancel the common factor.
Step 5.3.3.1.2.3
Rewrite the expression.
Step 5.3.3.1.3
Apply the distributive property.
Step 5.3.3.1.4
Cancel the common factor of .
Step 5.3.3.1.4.1
Factor out of .
Step 5.3.3.1.4.2
Cancel the common factor.
Step 5.3.3.1.4.3
Rewrite the expression.
Step 5.3.3.1.5
Combine and .
Step 5.3.3.1.6
To write as a fraction with a common denominator, multiply by .
Step 5.3.3.1.7
Combine and .
Step 5.3.3.1.8
Combine the numerators over the common denominator.
Step 5.3.3.1.9
Move to the left of .
Step 5.3.3.2
Combine the numerators over the common denominator.
Step 5.3.3.3
Combine the opposite terms in .
Step 5.3.3.3.1
Subtract from .
Step 5.3.3.3.2
Add and .
Step 5.3.3.4
Cancel the common factor of .
Step 5.3.3.4.1
Cancel the common factor.
Step 5.3.3.4.2
Divide by .
Step 5.3.3.5
The functions sine and arcsine are inverses.
Step 5.3.3.6
Cancel the common factor of .
Step 5.3.3.6.1
Cancel the common factor.
Step 5.3.3.6.2
Rewrite the expression.
Step 5.3.4
Combine the opposite terms in .
Step 5.3.4.1
Add and .
Step 5.3.4.2
Add and .
Step 5.4
Since and , then is the inverse of .