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Trigonometry Examples
Step 1
Interchange the variables.
Step 2
Step 2.1
Rewrite the equation as .
Step 2.2
Divide each term in the equation by .
Step 2.3
Separate fractions.
Step 2.4
Convert from to .
Step 2.5
Divide by .
Step 2.6
Apply the sine double-angle identity.
Step 2.7
Combine fractions.
Step 2.7.1
Combine and .
Step 2.7.2
Reorder factors in .
Step 2.8
Separate fractions.
Step 2.9
Convert from to .
Step 2.10
Divide by .
Step 2.11
Simplify the left side.
Step 2.11.1
Simplify .
Step 2.11.1.1
Rewrite in terms of sines and cosines.
Step 2.11.1.2
Combine and .
Step 2.11.1.3
Multiply the numerator by the reciprocal of the denominator.
Step 2.11.1.4
Multiply by .
Step 2.12
Simplify the right side.
Step 2.12.1
Simplify .
Step 2.12.1.1
Rewrite in terms of sines and cosines.
Step 2.12.1.2
Combine and .
Step 2.13
Multiply both sides of the equation by .
Step 2.14
Cancel the common factor of .
Step 2.14.1
Cancel the common factor.
Step 2.14.2
Rewrite the expression.
Step 2.15
Cancel the common factor of .
Step 2.15.1
Cancel the common factor.
Step 2.15.2
Rewrite the expression.
Step 2.16
Multiply both sides by .
Step 2.17
Simplify.
Step 2.17.1
Simplify the left side.
Step 2.17.1.1
Cancel the common factor of .
Step 2.17.1.1.1
Cancel the common factor.
Step 2.17.1.1.2
Rewrite the expression.
Step 2.17.2
Simplify the right side.
Step 2.17.2.1
Simplify .
Step 2.17.2.1.1
Apply the distributive property.
Step 2.17.2.1.2
Multiply by .
Step 2.18
Solve for .
Step 2.18.1
Simplify the left side.
Step 2.18.1.1
Simplify .
Step 2.18.1.1.1
Rewrite.
Step 2.18.1.1.2
Rewrite as .
Step 2.18.1.1.3
Expand using the FOIL Method.
Step 2.18.1.1.3.1
Apply the distributive property.
Step 2.18.1.1.3.2
Apply the distributive property.
Step 2.18.1.1.3.3
Apply the distributive property.
Step 2.18.1.1.4
Simplify and combine like terms.
Step 2.18.1.1.4.1
Simplify each term.
Step 2.18.1.1.4.1.1
Multiply .
Step 2.18.1.1.4.1.1.1
Raise to the power of .
Step 2.18.1.1.4.1.1.2
Raise to the power of .
Step 2.18.1.1.4.1.1.3
Use the power rule to combine exponents.
Step 2.18.1.1.4.1.1.4
Add and .
Step 2.18.1.1.4.1.2
Multiply .
Step 2.18.1.1.4.1.2.1
Raise to the power of .
Step 2.18.1.1.4.1.2.2
Raise to the power of .
Step 2.18.1.1.4.1.2.3
Use the power rule to combine exponents.
Step 2.18.1.1.4.1.2.4
Add and .
Step 2.18.1.1.4.2
Reorder the factors of .
Step 2.18.1.1.4.3
Add and .
Step 2.18.1.1.5
Move .
Step 2.18.1.1.6
Apply pythagorean identity.
Step 2.18.1.1.7
Simplify each term.
Step 2.18.1.1.7.1
Reorder and .
Step 2.18.1.1.7.2
Reorder and .
Step 2.18.1.1.7.3
Apply the sine double-angle identity.
Step 2.18.2
Substitute for .
Step 2.18.3
Subtract from both sides of the equation.
Step 2.18.4
Subtract from both sides of the equation.
Step 2.18.5
Factor out of .
Step 2.18.5.1
Factor out of .
Step 2.18.5.2
Factor out of .
Step 2.18.5.3
Factor out of .
Step 2.18.6
Divide each term in by and simplify.
Step 2.18.6.1
Divide each term in by .
Step 2.18.6.2
Simplify the left side.
Step 2.18.6.2.1
Cancel the common factor of .
Step 2.18.6.2.1.1
Cancel the common factor.
Step 2.18.6.2.1.2
Divide by .
Step 2.18.6.3
Simplify the right side.
Step 2.18.6.3.1
Combine the numerators over the common denominator.
Step 2.18.6.3.2
Cancel the common factor of and .
Step 2.18.6.3.2.1
Factor out of .
Step 2.18.6.3.2.2
Rewrite as .
Step 2.18.6.3.2.3
Factor out of .
Step 2.18.6.3.2.4
Reorder terms.
Step 2.18.6.3.2.5
Cancel the common factor.
Step 2.18.6.3.2.6
Divide by .
Step 2.18.7
Substitute for .
Step 2.18.8
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 2.18.9
Simplify the right side.
Step 2.18.9.1
The exact value of is .
Step 2.18.10
Divide each term in by and simplify.
Step 2.18.10.1
Divide each term in by .
Step 2.18.10.2
Simplify the left side.
Step 2.18.10.2.1
Cancel the common factor of .
Step 2.18.10.2.1.1
Cancel the common factor.
Step 2.18.10.2.1.2
Divide by .
Step 2.18.10.3
Simplify the right side.
Step 2.18.10.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 2.18.10.3.2
Multiply .
Step 2.18.10.3.2.1
Multiply by .
Step 2.18.10.3.2.2
Multiply by .
Step 2.18.11
The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from , to find a reference angle. Next, add this reference angle to to find the solution in the third quadrant.
Step 2.18.12
Simplify the expression to find the second solution.
Step 2.18.12.1
Subtract from .
Step 2.18.12.2
The resulting angle of is positive, less than , and coterminal with .
Step 2.18.12.3
Divide each term in by and simplify.
Step 2.18.12.3.1
Divide each term in by .
Step 2.18.12.3.2
Simplify the left side.
Step 2.18.12.3.2.1
Cancel the common factor of .
Step 2.18.12.3.2.1.1
Cancel the common factor.
Step 2.18.12.3.2.1.2
Divide by .
Step 2.18.12.3.3
Simplify the right side.
Step 2.18.12.3.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 2.18.12.3.3.2
Multiply .
Step 2.18.12.3.3.2.1
Multiply by .
Step 2.18.12.3.3.2.2
Multiply by .
Step 2.18.13
Find the period of .
Step 2.18.13.1
The period of the function can be calculated using .
Step 2.18.13.2
Replace with in the formula for period.
Step 2.18.13.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 2.18.13.4
Cancel the common factor of .
Step 2.18.13.4.1
Cancel the common factor.
Step 2.18.13.4.2
Divide by .
Step 2.18.14
Add to every negative angle to get positive angles.
Step 2.18.14.1
Add to to find the positive angle.
Step 2.18.14.2
To write as a fraction with a common denominator, multiply by .
Step 2.18.14.3
Combine fractions.
Step 2.18.14.3.1
Combine and .
Step 2.18.14.3.2
Combine the numerators over the common denominator.
Step 2.18.14.4
Simplify the numerator.
Step 2.18.14.4.1
Move to the left of .
Step 2.18.14.4.2
Subtract from .
Step 2.18.14.5
List the new angles.
Step 2.18.15
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
, for any integer
Step 3
Replace with to show the final answer.
Step 4
Step 4.1
The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of and and compare them.
Step 4.2
Find the range of .
Step 4.2.1
The range is the set of all valid values. Use the graph to find the range.
Interval Notation:
Step 4.3
Find the domain of .
Step 4.3.1
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
Step 4.4
Since the domain of is not equal to the range of , then is not an inverse of .
There is no inverse
There is no inverse
Step 5