Trigonometry Examples

Find the Inverse (sin(30))/x=(sin(60))/y
Step 1
Rewrite the equation as .
Step 2
Factor each term.
Tap for more steps...
Step 2.1
The exact value of is .
Step 2.2
Multiply the numerator by the reciprocal of the denominator.
Step 2.3
Multiply by .
Step 2.4
The exact value of is .
Step 2.5
Multiply the numerator by the reciprocal of the denominator.
Step 2.6
Multiply by .
Step 3
Find the LCD of the terms in the equation.
Tap for more steps...
Step 3.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 3.2
Since contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part then find LCM for the variable part .
Step 3.3
The LCM is the smallest positive number that all of the numbers divide into evenly.
1. List the prime factors of each number.
2. Multiply each factor the greatest number of times it occurs in either number.
Step 3.4
Since has no factors besides and .
is a prime number
Step 3.5
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.
Step 3.6
The factor for is itself.
occurs time.
Step 3.7
The factor for is itself.
occurs time.
Step 3.8
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either term.
Step 3.9
Multiply by .
Step 3.10
The LCM for is the numeric part multiplied by the variable part.
Step 4
Multiply each term in by to eliminate the fractions.
Tap for more steps...
Step 4.1
Multiply each term in by .
Step 4.2
Simplify the left side.
Tap for more steps...
Step 4.2.1
Rewrite using the commutative property of multiplication.
Step 4.2.2
Cancel the common factor of .
Tap for more steps...
Step 4.2.2.1
Factor out of .
Step 4.2.2.2
Cancel the common factor.
Step 4.2.2.3
Rewrite the expression.
Step 4.2.3
Cancel the common factor of .
Tap for more steps...
Step 4.2.3.1
Factor out of .
Step 4.2.3.2
Cancel the common factor.
Step 4.2.3.3
Rewrite the expression.
Step 4.3
Simplify the right side.
Tap for more steps...
Step 4.3.1
Rewrite using the commutative property of multiplication.
Step 4.3.2
Cancel the common factor of .
Tap for more steps...
Step 4.3.2.1
Factor out of .
Step 4.3.2.2
Cancel the common factor.
Step 4.3.2.3
Rewrite the expression.
Step 4.3.3
Cancel the common factor of .
Tap for more steps...
Step 4.3.3.1
Factor out of .
Step 4.3.3.2
Cancel the common factor.
Step 4.3.3.3
Rewrite the expression.
Step 5
Rewrite the equation as .
Step 6
Interchange the variables.
Step 7
Solve for .
Tap for more steps...
Step 7.1
Rewrite the equation as .
Step 7.2
Divide each term in by and simplify.
Tap for more steps...
Step 7.2.1
Divide each term in by .
Step 7.2.2
Simplify the left side.
Tap for more steps...
Step 7.2.2.1
Cancel the common factor of .
Tap for more steps...
Step 7.2.2.1.1
Cancel the common factor.
Step 7.2.2.1.2
Divide by .
Step 7.2.3
Simplify the right side.
Tap for more steps...
Step 7.2.3.1
Multiply by .
Step 7.2.3.2
Combine and simplify the denominator.
Tap for more steps...
Step 7.2.3.2.1
Multiply by .
Step 7.2.3.2.2
Raise to the power of .
Step 7.2.3.2.3
Raise to the power of .
Step 7.2.3.2.4
Use the power rule to combine exponents.
Step 7.2.3.2.5
Add and .
Step 7.2.3.2.6
Rewrite as .
Tap for more steps...
Step 7.2.3.2.6.1
Use to rewrite as .
Step 7.2.3.2.6.2
Apply the power rule and multiply exponents, .
Step 7.2.3.2.6.3
Combine and .
Step 7.2.3.2.6.4
Cancel the common factor of .
Tap for more steps...
Step 7.2.3.2.6.4.1
Cancel the common factor.
Step 7.2.3.2.6.4.2
Rewrite the expression.
Step 7.2.3.2.6.5
Evaluate the exponent.
Step 8
Replace with to show the final answer.
Step 9
Verify if is the inverse of .
Tap for more steps...
Step 9.1
To verify the inverse, check if and .
Step 9.2
Evaluate .
Tap for more steps...
Step 9.2.1
Set up the composite result function.
Step 9.2.2
Evaluate by substituting in the value of into .
Step 9.2.3
Simplify the numerator.
Tap for more steps...
Step 9.2.3.1
Raise to the power of .
Step 9.2.3.2
Raise to the power of .
Step 9.2.3.3
Use the power rule to combine exponents.
Step 9.2.3.4
Add and .
Step 9.2.4
Simplify terms.
Tap for more steps...
Step 9.2.4.1
Rewrite as .
Tap for more steps...
Step 9.2.4.1.1
Use to rewrite as .
Step 9.2.4.1.2
Apply the power rule and multiply exponents, .
Step 9.2.4.1.3
Combine and .
Step 9.2.4.1.4
Cancel the common factor of .
Tap for more steps...
Step 9.2.4.1.4.1
Cancel the common factor.
Step 9.2.4.1.4.2
Rewrite the expression.
Step 9.2.4.1.5
Evaluate the exponent.
Step 9.2.4.2
Cancel the common factor of .
Tap for more steps...
Step 9.2.4.2.1
Cancel the common factor.
Step 9.2.4.2.2
Divide by .
Step 9.3
Evaluate .
Tap for more steps...
Step 9.3.1
Set up the composite result function.
Step 9.3.2
Evaluate by substituting in the value of into .
Step 9.3.3
Multiply .
Tap for more steps...
Step 9.3.3.1
Combine and .
Step 9.3.3.2
Raise to the power of .
Step 9.3.3.3
Raise to the power of .
Step 9.3.3.4
Use the power rule to combine exponents.
Step 9.3.3.5
Add and .
Step 9.3.4
Rewrite as .
Tap for more steps...
Step 9.3.4.1
Use to rewrite as .
Step 9.3.4.2
Apply the power rule and multiply exponents, .
Step 9.3.4.3
Combine and .
Step 9.3.4.4
Cancel the common factor of .
Tap for more steps...
Step 9.3.4.4.1
Cancel the common factor.
Step 9.3.4.4.2
Rewrite the expression.
Step 9.3.4.5
Evaluate the exponent.
Step 9.3.5
Cancel the common factor of .
Tap for more steps...
Step 9.3.5.1
Cancel the common factor.
Step 9.3.5.2
Divide by .
Step 9.4
Since and , then is the inverse of .