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Trigonometry Examples
Step 1
Divide each term in the equation by .
Step 2
Replace with an equivalent expression in the numerator.
Step 3
Apply the distributive property.
Step 4
Step 4.1
Multiply by .
Step 4.2
Multiply by .
Step 5
Rewrite in terms of sines and cosines.
Step 6
Step 6.1
Apply the distributive property.
Step 6.2
Combine and .
Step 6.3
Cancel the common factor of .
Step 6.3.1
Factor out of .
Step 6.3.2
Cancel the common factor.
Step 6.3.3
Rewrite the expression.
Step 7
Step 7.1
Separate fractions.
Step 7.2
Convert from to .
Step 7.3
Divide by .
Step 8
Factor out of .
Step 9
Separate fractions.
Step 10
Convert from to .
Step 11
Divide by .
Step 12
Step 12.1
Simplify each term.
Step 12.1.1
Rewrite in terms of sines and cosines.
Step 12.1.2
Combine and .
Step 13
Step 13.1
Simplify .
Step 13.1.1
Rewrite in terms of sines and cosines.
Step 13.1.2
Multiply .
Step 13.1.2.1
Combine and .
Step 13.1.2.2
Raise to the power of .
Step 13.1.2.3
Raise to the power of .
Step 13.1.2.4
Use the power rule to combine exponents.
Step 13.1.2.5
Add and .
Step 14
Multiply both sides of the equation by .
Step 15
Apply the distributive property.
Step 16
Step 16.1
Cancel the common factor.
Step 16.2
Rewrite the expression.
Step 17
Move to the left of .
Step 18
Step 18.1
Cancel the common factor.
Step 18.2
Rewrite the expression.
Step 19
Subtract from both sides of the equation.
Step 20
Replace with .
Step 21
Step 21.1
Substitute for .
Step 21.2
Simplify .
Step 21.2.1
Simplify each term.
Step 21.2.1.1
Apply the distributive property.
Step 21.2.1.2
Multiply by .
Step 21.2.1.3
Multiply .
Step 21.2.1.3.1
Multiply by .
Step 21.2.1.3.2
Multiply by .
Step 21.2.2
Subtract from .
Step 21.3
Factor using the AC method.
Step 21.3.1
Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Step 21.3.2
Write the factored form using these integers.
Step 21.4
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 21.5
Set equal to and solve for .
Step 21.5.1
Set equal to .
Step 21.5.2
Add to both sides of the equation.
Step 21.6
Set equal to and solve for .
Step 21.6.1
Set equal to .
Step 21.6.2
Add to both sides of the equation.
Step 21.7
The final solution is all the values that make true.
Step 21.8
Substitute for .
Step 21.9
Set up each of the solutions to solve for .
Step 21.10
Solve for in .
Step 21.10.1
The range of cosine is . Since does not fall in this range, there is no solution.
No solution
No solution
Step 21.11
Solve for in .
Step 21.11.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 21.11.2
Simplify the right side.
Step 21.11.2.1
The exact value of is .
Step 21.11.3
The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from to find the solution in the fourth quadrant.
Step 21.11.4
Subtract from .
Step 21.11.5
Find the period of .
Step 21.11.5.1
The period of the function can be calculated using .
Step 21.11.5.2
Replace with in the formula for period.
Step 21.11.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 21.11.5.4
Divide by .
Step 21.11.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 21.12
List all of the solutions.
, for any integer
Step 21.13
Consolidate the answers.
, for any integer
, for any integer