Enter a problem...
Trigonometry Examples
Step 1
Interchange the variables.
Step 2
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.
Step 2.4.1
Simplify .
Step 2.4.1.1
Write the expression using exponents.
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
Find the LCD of the terms in the equation.
Step 2.5.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 2.5.2
The LCM of one and any expression is the expression.
Step 2.6
Multiply each term in by to eliminate the fractions.
Step 2.6.1
Multiply each term in by .
Step 2.6.2
Simplify the left side.
Step 2.6.2.1
Cancel the common factor of .
Step 2.6.2.1.1
Cancel the common factor.
Step 2.6.2.1.2
Rewrite the expression.
Step 2.7
Solve the equation.
Step 2.7.1
Rewrite the equation as .
Step 2.7.2
Divide each term in by and simplify.
Step 2.7.2.1
Divide each term in by .
Step 2.7.2.2
Simplify the left side.
Step 2.7.2.2.1
Cancel the common factor of .
Step 2.7.2.2.1.1
Cancel the common factor.
Step 2.7.2.2.1.2
Divide by .
Step 2.7.2.3
Simplify the right side.
Step 2.7.2.3.1
Multiply by .
Step 2.7.2.3.2
Combine and simplify the denominator.
Step 2.7.2.3.2.1
Multiply by .
Step 2.7.2.3.2.2
Raise to the power of .
Step 2.7.2.3.2.3
Raise to the power of .
Step 2.7.2.3.2.4
Use the power rule to combine exponents.
Step 2.7.2.3.2.5
Add and .
Step 2.7.2.3.2.6
Rewrite as .
Step 2.7.2.3.2.6.1
Use to rewrite as .
Step 2.7.2.3.2.6.2
Apply the power rule and multiply exponents, .
Step 2.7.2.3.2.6.3
Combine and .
Step 2.7.2.3.2.6.4
Cancel the common factor of .
Step 2.7.2.3.2.6.4.1
Cancel the common factor.
Step 2.7.2.3.2.6.4.2
Rewrite the expression.
Step 2.7.2.3.2.6.5
Simplify.
Step 3
Replace with to show the final answer.
Step 4
Step 4.1
To verify the inverse, check if and .
Step 4.2
Evaluate .
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.
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
Write as a fraction with a common denominator.
Step 4.2.4.5
Combine the numerators over the common denominator.
Step 4.2.4.6
Write as a fraction with a common denominator.
Step 4.2.4.7
Combine the numerators over the common denominator.
Step 4.2.4.8
Multiply by .
Step 4.2.4.9
Multiply by .
Step 4.2.4.10
Rewrite as .
Step 4.2.4.10.1
Factor the perfect power out of .
Step 4.2.4.10.2
Factor the perfect power out of .
Step 4.2.4.10.3
Rearrange the fraction .
Step 4.2.4.11
Pull terms out from under the radical.
Step 4.2.4.12
Combine and .
Step 4.2.4.13
Write as a fraction with a common denominator.
Step 4.2.4.14
Combine the numerators over the common denominator.
Step 4.2.4.15
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 4.2.4.16
Rewrite as .
Step 4.2.4.17
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.2.4.18
Write as a fraction with a common denominator.
Step 4.2.4.19
Combine the numerators over the common denominator.
Step 4.2.4.20
Write as a fraction with a common denominator.
Step 4.2.4.21
Combine the numerators over the common denominator.
Step 4.2.4.22
Multiply by .
Step 4.2.4.23
Multiply by .
Step 4.2.4.24
Rewrite as .
Step 4.2.4.24.1
Factor the perfect power out of .
Step 4.2.4.24.2
Factor the perfect power out of .
Step 4.2.4.24.3
Rearrange the fraction .
Step 4.2.4.25
Pull terms out from under the radical.
Step 4.2.4.26
Combine and .
Step 4.2.4.27
Write as a fraction with a common denominator.
Step 4.2.4.28
Combine the numerators over the common denominator.
Step 4.2.4.29
Multiply by .
Step 4.2.4.30
Multiply by .
Step 4.2.4.31
Expand using the FOIL Method.
Step 4.2.4.31.1
Apply the distributive property.
Step 4.2.4.31.2
Apply the distributive property.
Step 4.2.4.31.3
Apply the distributive property.
Step 4.2.4.32
Combine the opposite terms in .
Step 4.2.4.32.1
Reorder the factors in the terms and .
Step 4.2.4.32.2
Add and .
Step 4.2.4.32.3
Add and .
Step 4.2.4.33
Simplify each term.
Step 4.2.4.33.1
Multiply by .
Step 4.2.4.33.2
Rewrite using the commutative property of multiplication.
Step 4.2.4.33.3
Multiply .
Step 4.2.4.33.3.1
Raise to the power of .
Step 4.2.4.33.3.2
Raise to the power of .
Step 4.2.4.33.3.3
Use the power rule to combine exponents.
Step 4.2.4.33.3.4
Add and .
Step 4.2.4.33.4
Rewrite as .
Step 4.2.4.33.4.1
Use to rewrite as .
Step 4.2.4.33.4.2
Apply the power rule and multiply exponents, .
Step 4.2.4.33.4.3
Combine and .
Step 4.2.4.33.4.4
Cancel the common factor of .
Step 4.2.4.33.4.4.1
Cancel the common factor.
Step 4.2.4.33.4.4.2
Rewrite the expression.
Step 4.2.4.33.4.5
Simplify.
Step 4.2.4.33.5
Expand using the FOIL Method.
Step 4.2.4.33.5.1
Apply the distributive property.
Step 4.2.4.33.5.2
Apply the distributive property.
Step 4.2.4.33.5.3
Apply the distributive property.
Step 4.2.4.33.6
Combine the opposite terms in .
Step 4.2.4.33.6.1
Reorder the factors in the terms and .
Step 4.2.4.33.6.2
Add and .
Step 4.2.4.33.6.3
Add and .
Step 4.2.4.33.7
Simplify each term.
Step 4.2.4.33.7.1
Multiply by .
Step 4.2.4.33.7.2
Multiply by .
Step 4.2.4.33.8
Apply the distributive property.
Step 4.2.4.33.9
Multiply by .
Step 4.2.4.34
Subtract from .
Step 4.2.4.35
Add and .
Step 4.2.4.36
Rewrite as .
Step 4.2.4.37
Rewrite as .
Step 4.2.4.38
Pull terms out from under the radical, assuming positive real numbers.
Step 4.2.5
Simplify the denominator.
Step 4.2.5.1
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 4.2.5.2
Rewrite as .
Step 4.2.5.3
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.2.5.4
Write as a fraction with a common denominator.
Step 4.2.5.5
Combine the numerators over the common denominator.
Step 4.2.5.6
Write as a fraction with a common denominator.
Step 4.2.5.7
Combine the numerators over the common denominator.
Step 4.2.5.8
Multiply by .
Step 4.2.5.9
Multiply by .
Step 4.2.5.10
Rewrite as .
Step 4.2.5.10.1
Factor the perfect power out of .
Step 4.2.5.10.2
Factor the perfect power out of .
Step 4.2.5.10.3
Rearrange the fraction .
Step 4.2.5.11
Pull terms out from under the radical.
Step 4.2.5.12
Combine and .
Step 4.2.5.13
Write as a fraction with a common denominator.
Step 4.2.5.14
Combine the numerators over the common denominator.
Step 4.2.5.15
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 4.2.5.16
Rewrite as .
Step 4.2.5.17
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.2.5.18
Write as a fraction with a common denominator.
Step 4.2.5.19
Combine the numerators over the common denominator.
Step 4.2.5.20
Write as a fraction with a common denominator.
Step 4.2.5.21
Combine the numerators over the common denominator.
Step 4.2.5.22
Multiply by .
Step 4.2.5.23
Multiply by .
Step 4.2.5.24
Rewrite as .
Step 4.2.5.24.1
Factor the perfect power out of .
Step 4.2.5.24.2
Factor the perfect power out of .
Step 4.2.5.24.3
Rearrange the fraction .
Step 4.2.5.25
Pull terms out from under the radical.
Step 4.2.5.26
Combine and .
Step 4.2.5.27
Write as a fraction with a common denominator.
Step 4.2.5.28
Combine the numerators over the common denominator.
Step 4.2.6
Combine fractions.
Step 4.2.6.1
Combine and .
Step 4.2.6.2
Multiply by .
Step 4.2.6.3
Multiply by .
Step 4.2.7
Simplify the denominator.
Step 4.2.7.1
Use the power rule to combine exponents.
Step 4.2.7.2
Add and .
Step 4.2.8
Simplify the denominator.
Step 4.2.8.1
Expand using the FOIL Method.
Step 4.2.8.1.1
Apply the distributive property.
Step 4.2.8.1.2
Apply the distributive property.
Step 4.2.8.1.3
Apply the distributive property.
Step 4.2.8.2
Combine the opposite terms in .
Step 4.2.8.2.1
Reorder the factors in the terms and .
Step 4.2.8.2.2
Add and .
Step 4.2.8.2.3
Add and .
Step 4.2.8.3
Simplify each term.
Step 4.2.8.3.1
Multiply by .
Step 4.2.8.3.2
Rewrite using the commutative property of multiplication.
Step 4.2.8.3.3
Multiply .
Step 4.2.8.3.3.1
Raise to the power of .
Step 4.2.8.3.3.2
Raise to the power of .
Step 4.2.8.3.3.3
Use the power rule to combine exponents.
Step 4.2.8.3.3.4
Add and .
Step 4.2.8.3.4
Rewrite as .
Step 4.2.8.3.4.1
Use to rewrite as .
Step 4.2.8.3.4.2
Apply the power rule and multiply exponents, .
Step 4.2.8.3.4.3
Combine and .
Step 4.2.8.3.4.4
Cancel the common factor of .
Step 4.2.8.3.4.4.1
Cancel the common factor.
Step 4.2.8.3.4.4.2
Rewrite the expression.
Step 4.2.8.3.4.5
Simplify.
Step 4.2.8.3.5
Expand using the FOIL Method.
Step 4.2.8.3.5.1
Apply the distributive property.
Step 4.2.8.3.5.2
Apply the distributive property.
Step 4.2.8.3.5.3
Apply the distributive property.
Step 4.2.8.3.6
Combine the opposite terms in .
Step 4.2.8.3.6.1
Reorder the factors in the terms and .
Step 4.2.8.3.6.2
Add and .
Step 4.2.8.3.6.3
Add and .
Step 4.2.8.3.7
Simplify each term.
Step 4.2.8.3.7.1
Multiply by .
Step 4.2.8.3.7.2
Multiply by .
Step 4.2.8.3.8
Apply the distributive property.
Step 4.2.8.3.9
Multiply by .
Step 4.2.8.4
Subtract from .
Step 4.2.8.5
Add and .
Step 4.2.9
Multiply the numerator by the reciprocal of the denominator.
Step 4.2.10
Combine.
Step 4.2.11
Cancel the common factor of .
Step 4.2.11.1
Cancel the common factor.
Step 4.2.11.2
Rewrite the expression.
Step 4.2.12
Cancel the common factor of and .
Step 4.2.12.1
Factor out of .
Step 4.2.12.2
Cancel the common factors.
Step 4.2.12.2.1
Raise to the power of .
Step 4.2.12.2.2
Factor out of .
Step 4.2.12.2.3
Cancel the common factor.
Step 4.2.12.2.4
Rewrite the expression.
Step 4.2.12.2.5
Divide by .
Step 4.3
Evaluate .
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
Write the expression using exponents.
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.
Step 4.3.5.1
Multiply the numerator by the reciprocal of the denominator.
Step 4.3.5.2
Cancel the common factor of .
Step 4.3.5.2.1
Factor out of .
Step 4.3.5.2.2
Cancel the common factor.
Step 4.3.5.2.3
Rewrite the expression.
Step 4.3.5.3
Multiply by .
Step 4.3.5.4
Combine and simplify the denominator.
Step 4.3.5.4.1
Multiply by .
Step 4.3.5.4.2
Raise to the power of .
Step 4.3.5.4.3
Raise to the power of .
Step 4.3.5.4.4
Use the power rule to combine exponents.
Step 4.3.5.4.5
Add and .
Step 4.3.5.4.6
Rewrite as .
Step 4.3.5.4.6.1
Use to rewrite as .
Step 4.3.5.4.6.2
Apply the power rule and multiply exponents, .
Step 4.3.5.4.6.3
Combine and .
Step 4.3.5.4.6.4
Cancel the common factor of .
Step 4.3.5.4.6.4.1
Cancel the common factor.
Step 4.3.5.4.6.4.2
Rewrite the expression.
Step 4.3.5.4.6.5
Simplify.
Step 4.3.5.5
Cancel the common factor of .
Step 4.3.5.5.1
Cancel the common factor.
Step 4.3.5.5.2
Rewrite the expression.
Step 4.3.5.6
Cancel the common factor of .
Step 4.3.5.6.1
Cancel the common factor.
Step 4.3.5.6.2
Divide by .
Step 4.3.6
Simplify each term.
Step 4.3.6.1
Multiply the numerator by the reciprocal of the denominator.
Step 4.3.6.2
Cancel the common factor of .
Step 4.3.6.2.1
Factor out of .
Step 4.3.6.2.2
Cancel the common factor.
Step 4.3.6.2.3
Rewrite the expression.
Step 4.3.6.3
Multiply by .
Step 4.3.6.4
Combine and simplify the denominator.
Step 4.3.6.4.1
Multiply by .
Step 4.3.6.4.2
Raise to the power of .
Step 4.3.6.4.3
Raise to the power of .
Step 4.3.6.4.4
Use the power rule to combine exponents.
Step 4.3.6.4.5
Add and .
Step 4.3.6.4.6
Rewrite as .
Step 4.3.6.4.6.1
Use to rewrite as .
Step 4.3.6.4.6.2
Apply the power rule and multiply exponents, .
Step 4.3.6.4.6.3
Combine and .
Step 4.3.6.4.6.4
Cancel the common factor of .
Step 4.3.6.4.6.4.1
Cancel the common factor.
Step 4.3.6.4.6.4.2
Rewrite the expression.
Step 4.3.6.4.6.5
Simplify.
Step 4.3.6.5
Cancel the common factor of .
Step 4.3.6.5.1
Cancel the common factor.
Step 4.3.6.5.2
Rewrite the expression.
Step 4.3.6.6
Cancel the common factor of .
Step 4.3.6.6.1
Cancel the common factor.
Step 4.3.6.6.2
Divide by .
Step 4.4
Since and , then is the inverse of .