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Trigonometry Examples
Step 1
Interchange the variables.
Step 2
Step 2.1
Rewrite the equation as .
Step 2.2
Simplify the left side.
Step 2.2.1
Simplify .
Step 2.2.1.1
Simplify the numerator.
Step 2.2.1.1.1
Rewrite in terms of sines and cosines, then cancel the common factors.
Step 2.2.1.1.1.1
Rewrite in terms of sines and cosines.
Step 2.2.1.1.1.2
Cancel the common factors.
Step 2.2.1.1.2
Convert from to .
Step 2.2.1.1.3
Add and .
Step 2.2.1.2
Separate fractions.
Step 2.2.1.3
Rewrite in terms of sines and cosines.
Step 2.2.1.4
Rewrite in terms of sines and cosines.
Step 2.2.1.5
Multiply by the reciprocal of the fraction to divide by .
Step 2.2.1.6
Cancel the common factor of .
Step 2.2.1.6.1
Cancel the common factor.
Step 2.2.1.6.2
Rewrite the expression.
Step 2.2.1.7
Convert from to .
Step 2.2.1.8
Divide by .
Step 2.3
Divide each term in by and simplify.
Step 2.3.1
Divide each term in by .
Step 2.3.2
Simplify the left side.
Step 2.3.2.1
Cancel the common factor of .
Step 2.3.2.1.1
Cancel the common factor.
Step 2.3.2.1.2
Divide by .
Step 2.4
Take the inverse cosecant of both sides of the equation to extract from inside the cosecant.
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
Multiply the numerator by the reciprocal of the denominator.
Step 4.2.4
Simplify the numerator.
Step 4.2.4.1
Rewrite in terms of sines and cosines, then cancel the common factors.
Step 4.2.4.1.1
Rewrite in terms of sines and cosines.
Step 4.2.4.1.2
Cancel the common factors.
Step 4.2.4.2
Convert from to .
Step 4.2.4.3
Add and .
Step 4.2.5
Cancel the common factor of .
Step 4.2.5.1
Factor out of .
Step 4.2.5.2
Cancel the common factor.
Step 4.2.5.3
Rewrite the expression.
Step 4.2.6
Rewrite in terms of sines and cosines.
Step 4.2.7
Rewrite in terms of sines and cosines.
Step 4.2.8
Multiply by the reciprocal of the fraction to divide by .
Step 4.2.9
Cancel the common factor of .
Step 4.2.9.1
Cancel the common factor.
Step 4.2.9.2
Rewrite the expression.
Step 4.2.10
Convert from to .
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
Simplify the numerator.
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
Multiply the numerator by the reciprocal of the denominator.
Step 4.3.3.3
Combine.
Step 4.3.3.4
Multiply by .
Step 4.3.3.5
Simplify the denominator.
Step 4.3.3.5.1
Rewrite as .
Step 4.3.3.5.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.3.3.5.3
Simplify.
Step 4.3.3.5.3.1
Write as a fraction with a common denominator.
Step 4.3.3.5.3.2
Combine the numerators over the common denominator.
Step 4.3.3.5.3.3
To write as a fraction with a common denominator, multiply by .
Step 4.3.3.5.3.4
Combine and .
Step 4.3.3.5.3.5
Combine the numerators over the common denominator.
Step 4.3.3.5.3.6
Multiply by .
Step 4.3.3.5.4
Multiply by .
Step 4.3.3.5.5
Multiply by .
Step 4.3.3.5.6
Rewrite as .
Step 4.3.3.5.6.1
Factor the perfect power out of .
Step 4.3.3.5.6.2
Factor the perfect power out of .
Step 4.3.3.5.6.3
Rearrange the fraction .
Step 4.3.3.5.7
Pull terms out from under the radical.
Step 4.3.3.5.8
Combine and .
Step 4.3.3.6
Combine and .
Step 4.3.3.7
Reduce the expression by cancelling the common factors.
Step 4.3.3.7.1
Reduce the expression by cancelling the common factors.
Step 4.3.3.7.1.1
Cancel the common factor.
Step 4.3.3.7.1.2
Rewrite the expression.
Step 4.3.3.7.2
Divide by .
Step 4.3.3.8
Multiply by .
Step 4.3.3.9
Combine and simplify the denominator.
Step 4.3.3.9.1
Multiply by .
Step 4.3.3.9.2
Raise to the power of .
Step 4.3.3.9.3
Raise to the power of .
Step 4.3.3.9.4
Use the power rule to combine exponents.
Step 4.3.3.9.5
Add and .
Step 4.3.3.9.6
Rewrite as .
Step 4.3.3.9.6.1
Use to rewrite as .
Step 4.3.3.9.6.2
Apply the power rule and multiply exponents, .
Step 4.3.3.9.6.3
Combine and .
Step 4.3.3.9.6.4
Cancel the common factor of .
Step 4.3.3.9.6.4.1
Cancel the common factor.
Step 4.3.3.9.6.4.2
Rewrite the expression.
Step 4.3.3.9.6.5
Simplify.
Step 4.3.3.10
Rewrite in terms of sines and cosines, then cancel the common factors.
Step 4.3.3.10.1
Rewrite in terms of sines and cosines.
Step 4.3.3.10.2
Cancel the common factors.
Step 4.3.3.11
Convert from to .
Step 4.3.3.12
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 4.3.3.13
Multiply the numerator by the reciprocal of the denominator.
Step 4.3.3.14
Combine.
Step 4.3.3.15
Multiply by .
Step 4.3.3.16
Simplify the denominator.
Step 4.3.3.16.1
Rewrite as .
Step 4.3.3.16.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.3.3.16.3
Simplify.
Step 4.3.3.16.3.1
Write as a fraction with a common denominator.
Step 4.3.3.16.3.2
Combine the numerators over the common denominator.
Step 4.3.3.16.3.3
To write as a fraction with a common denominator, multiply by .
Step 4.3.3.16.3.4
Combine and .
Step 4.3.3.16.3.5
Combine the numerators over the common denominator.
Step 4.3.3.16.3.6
Multiply by .
Step 4.3.3.16.4
Multiply by .
Step 4.3.3.16.5
Multiply by .
Step 4.3.3.16.6
Rewrite as .
Step 4.3.3.16.6.1
Factor the perfect power out of .
Step 4.3.3.16.6.2
Factor the perfect power out of .
Step 4.3.3.16.6.3
Rearrange the fraction .
Step 4.3.3.16.7
Pull terms out from under the radical.
Step 4.3.3.16.8
Combine and .
Step 4.3.3.17
Combine and .
Step 4.3.3.18
Reduce the expression by cancelling the common factors.
Step 4.3.3.18.1
Reduce the expression by cancelling the common factors.
Step 4.3.3.18.1.1
Cancel the common factor.
Step 4.3.3.18.1.2
Rewrite the expression.
Step 4.3.3.18.2
Divide by .
Step 4.3.3.19
Multiply by .
Step 4.3.3.20
Combine and simplify the denominator.
Step 4.3.3.20.1
Multiply by .
Step 4.3.3.20.2
Raise to the power of .
Step 4.3.3.20.3
Raise to the power of .
Step 4.3.3.20.4
Use the power rule to combine exponents.
Step 4.3.3.20.5
Add and .
Step 4.3.3.20.6
Rewrite as .
Step 4.3.3.20.6.1
Use to rewrite as .
Step 4.3.3.20.6.2
Apply the power rule and multiply exponents, .
Step 4.3.3.20.6.3
Combine and .
Step 4.3.3.20.6.4
Cancel the common factor of .
Step 4.3.3.20.6.4.1
Cancel the common factor.
Step 4.3.3.20.6.4.2
Rewrite the expression.
Step 4.3.3.20.6.5
Simplify.
Step 4.3.3.21
Add and .
Step 4.3.4
Simplify the denominator.
Step 4.3.4.1
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 4.3.4.2
Simplify the denominator.
Step 4.3.4.2.1
Rewrite as .
Step 4.3.4.2.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.3.4.2.3
Simplify.
Step 4.3.4.2.3.1
Write as a fraction with a common denominator.
Step 4.3.4.2.3.2
Combine the numerators over the common denominator.
Step 4.3.4.2.3.3
To write as a fraction with a common denominator, multiply by .
Step 4.3.4.2.3.4
Combine and .
Step 4.3.4.2.3.5
Combine the numerators over the common denominator.
Step 4.3.4.2.3.6
Multiply by .
Step 4.3.4.2.4
Multiply by .
Step 4.3.4.2.5
Multiply by .
Step 4.3.4.2.6
Rewrite as .
Step 4.3.4.2.6.1
Factor the perfect power out of .
Step 4.3.4.2.6.2
Factor the perfect power out of .
Step 4.3.4.2.6.3
Rearrange the fraction .
Step 4.3.4.2.7
Pull terms out from under the radical.
Step 4.3.4.2.8
Combine and .
Step 4.3.4.3
Multiply the numerator by the reciprocal of the denominator.
Step 4.3.4.4
Multiply by .
Step 4.3.4.5
Multiply by .
Step 4.3.4.6
Combine and simplify the denominator.
Step 4.3.4.6.1
Multiply by .
Step 4.3.4.6.2
Raise to the power of .
Step 4.3.4.6.3
Raise to the power of .
Step 4.3.4.6.4
Use the power rule to combine exponents.
Step 4.3.4.6.5
Add and .
Step 4.3.4.6.6
Rewrite as .
Step 4.3.4.6.6.1
Use to rewrite as .
Step 4.3.4.6.6.2
Apply the power rule and multiply exponents, .
Step 4.3.4.6.6.3
Combine and .
Step 4.3.4.6.6.4
Cancel the common factor of .
Step 4.3.4.6.6.4.1
Cancel the common factor.
Step 4.3.4.6.6.4.2
Rewrite the expression.
Step 4.3.4.6.6.5
Simplify.
Step 4.3.5
Combine and .
Step 4.3.6
Multiply the numerator by the reciprocal of the denominator.
Step 4.3.7
Cancel the common factor of .
Step 4.3.7.1
Factor out of .
Step 4.3.7.2
Cancel the common factor.
Step 4.3.7.3
Rewrite the expression.
Step 4.3.8
Cancel the common factor of .
Step 4.3.8.1
Cancel the common factor.
Step 4.3.8.2
Rewrite the expression.
Step 4.4
Since and , then is the inverse of .