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Trigonometry Examples
Step 1
Divide each term in the equation by .
Step 2
Separate fractions.
Step 3
Convert from to .
Step 4
Divide by .
Step 5
Combine and .
Step 6
Separate fractions.
Step 7
Rewrite in terms of sines and cosines.
Step 8
Rewrite as a product.
Step 9
Step 9.1
Convert from to .
Step 9.2
Convert from to .
Step 10
Step 10.1
Divide by .
Step 10.2
Apply the distributive property.
Step 10.3
Multiply by .
Step 11
Step 11.1
Rewrite in terms of sines and cosines, then cancel the common factors.
Step 11.1.1
Add parentheses.
Step 11.1.2
Reorder and .
Step 11.1.3
Rewrite in terms of sines and cosines.
Step 11.1.4
Cancel the common factors.
Step 11.2
Multiply by .
Step 12
Separate fractions.
Step 13
Convert from to .
Step 14
Divide by .
Step 15
Evaluate .
Step 16
Combine and .
Step 17
Multiply both sides by .
Step 18
Step 18.1
Simplify the left side.
Step 18.1.1
Simplify .
Step 18.1.1.1
Simplify each term.
Step 18.1.1.1.1
Rewrite in terms of sines and cosines.
Step 18.1.1.1.2
Rewrite in terms of sines and cosines.
Step 18.1.1.1.3
Multiply by .
Step 18.1.1.1.4
Rewrite in terms of sines and cosines.
Step 18.1.1.2
Expand using the FOIL Method.
Step 18.1.1.2.1
Apply the distributive property.
Step 18.1.1.2.2
Apply the distributive property.
Step 18.1.1.2.3
Apply the distributive property.
Step 18.1.1.3
Simplify and combine like terms.
Step 18.1.1.3.1
Simplify each term.
Step 18.1.1.3.1.1
Multiply by .
Step 18.1.1.3.1.2
Cancel the common factor of .
Step 18.1.1.3.1.2.1
Factor out of .
Step 18.1.1.3.1.2.2
Cancel the common factor.
Step 18.1.1.3.1.2.3
Rewrite the expression.
Step 18.1.1.3.1.3
Multiply by .
Step 18.1.1.3.1.4
Combine and .
Step 18.1.1.3.2
Subtract from .
Step 18.1.1.3.3
Add and .
Step 18.1.1.4
Simplify each term.
Step 18.1.1.4.1
Separate fractions.
Step 18.1.1.4.2
Convert from to .
Step 18.1.1.4.3
Convert from to .
Step 18.1.1.4.4
Convert from to .
Step 18.2
Simplify the right side.
Step 18.2.1
Simplify .
Step 18.2.1.1
Cancel the common factor of .
Step 18.2.1.1.1
Cancel the common factor.
Step 18.2.1.1.2
Rewrite the expression.
Step 18.2.1.2
Rewrite in terms of sines and cosines.
Step 18.2.1.3
Combine and .
Step 18.2.1.4
Separate fractions.
Step 18.2.1.5
Convert from to .
Step 18.2.1.6
Divide by .
Step 19
Step 19.1
Simplify the left side.
Step 19.1.1
Simplify each term.
Step 19.1.1.1
Rewrite in terms of sines and cosines.
Step 19.1.1.2
Rewrite in terms of sines and cosines.
Step 19.1.1.3
Multiply by .
Step 19.1.1.4
Rewrite in terms of sines and cosines.
Step 19.2
Simplify the right side.
Step 19.2.1
Simplify .
Step 19.2.1.1
Rewrite in terms of sines and cosines.
Step 19.2.1.2
Combine and .
Step 19.3
Multiply both sides of the equation by .
Step 19.4
Apply the distributive property.
Step 19.5
Cancel the common factor of .
Step 19.5.1
Factor out of .
Step 19.5.2
Cancel the common factor.
Step 19.5.3
Rewrite the expression.
Step 19.6
Rewrite using the commutative property of multiplication.
Step 19.7
Cancel the common factor of .
Step 19.7.1
Factor out of .
Step 19.7.2
Cancel the common factor.
Step 19.7.3
Rewrite the expression.
Step 19.8
Combine and .
Step 19.9
Move to the left of .
Step 19.10
Multiply both sides of the equation by .
Step 19.11
Apply the distributive property.
Step 19.12
Cancel the common factor of .
Step 19.12.1
Cancel the common factor.
Step 19.12.2
Rewrite the expression.
Step 19.13
Rewrite using the commutative property of multiplication.
Step 19.14
Multiply .
Step 19.14.1
Raise to the power of .
Step 19.14.2
Raise to the power of .
Step 19.14.3
Use the power rule to combine exponents.
Step 19.14.4
Add and .
Step 19.15
Apply pythagorean identity.
Step 19.16
Cancel the common factor of .
Step 19.16.1
Cancel the common factor.
Step 19.16.2
Rewrite the expression.
Step 19.17
Subtract from both sides of the equation.
Step 19.18
Factor out of .
Step 19.18.1
Factor out of .
Step 19.18.2
Factor out of .
Step 19.18.3
Factor out of .
Step 19.19
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 19.20
Set equal to and solve for .
Step 19.20.1
Set equal to .
Step 19.20.2
Solve for .
Step 19.20.2.1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 19.20.2.2
Simplify the right side.
Step 19.20.2.2.1
The exact value of is .
Step 19.20.2.3
The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from to find the solution in the second quadrant.
Step 19.20.2.4
Subtract from .
Step 19.20.2.5
Find the period of .
Step 19.20.2.5.1
The period of the function can be calculated using .
Step 19.20.2.5.2
Replace with in the formula for period.
Step 19.20.2.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 19.20.2.5.4
Divide by .
Step 19.20.2.6
The period of the function is so values will repeat every degrees in both directions.
, for any integer
, for any integer
, for any integer
Step 19.21
Set equal to and solve for .
Step 19.21.1
Set equal to .
Step 19.21.2
Solve for .
Step 19.21.2.1
Add to both sides of the equation.
Step 19.21.2.2
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 19.21.2.3
Simplify the right side.
Step 19.21.2.3.1
Evaluate .
Step 19.21.2.4
The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from to find the solution in the second quadrant.
Step 19.21.2.5
Subtract from .
Step 19.21.2.6
Find the period of .
Step 19.21.2.6.1
The period of the function can be calculated using .
Step 19.21.2.6.2
Replace with in the formula for period.
Step 19.21.2.6.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 19.21.2.6.4
Divide by .
Step 19.21.2.7
The period of the function is so values will repeat every degrees in both directions.
, for any integer
, for any integer
, for any integer
Step 19.22
The final solution is all the values that make true.
, for any integer
, for any integer
Step 20
Consolidate and to .
, for any integer
Step 21
Exclude the solutions that do not make true.
, for any integer