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Trigonometry Examples
Step 1
Replace the with based on the identity.
Step 2
Step 2.1
Apply the distributive property.
Step 2.2
Multiply by .
Step 2.3
Multiply by .
Step 3
Subtract from .
Step 4
Substitute for .
Step 5
Step 5.1
Factor out of .
Step 5.1.1
Factor out of .
Step 5.1.2
Factor out of .
Step 5.1.3
Rewrite as .
Step 5.1.4
Factor out of .
Step 5.1.5
Factor out of .
Step 5.2
Factor.
Step 5.2.1
Factor by grouping.
Step 5.2.1.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Step 5.2.1.1.1
Factor out of .
Step 5.2.1.1.2
Rewrite as plus
Step 5.2.1.1.3
Apply the distributive property.
Step 5.2.1.1.4
Multiply by .
Step 5.2.1.2
Factor out the greatest common factor from each group.
Step 5.2.1.2.1
Group the first two terms and the last two terms.
Step 5.2.1.2.2
Factor out the greatest common factor (GCF) from each group.
Step 5.2.1.3
Factor the polynomial by factoring out the greatest common factor, .
Step 5.2.2
Remove unnecessary parentheses.
Step 6
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 7
Step 7.1
Set equal to .
Step 7.2
Solve for .
Step 7.2.1
Subtract from both sides of the equation.
Step 7.2.2
Divide each term in by and simplify.
Step 7.2.2.1
Divide each term in by .
Step 7.2.2.2
Simplify the left side.
Step 7.2.2.2.1
Cancel the common factor of .
Step 7.2.2.2.1.1
Cancel the common factor.
Step 7.2.2.2.1.2
Divide by .
Step 7.2.2.3
Simplify the right side.
Step 7.2.2.3.1
Move the negative in front of the fraction.
Step 8
Step 8.1
Set equal to .
Step 8.2
Subtract from both sides of the equation.
Step 9
The final solution is all the values that make true.
Step 10
Substitute for .
Step 11
Set up each of the solutions to solve for .
Step 12
Step 12.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 12.2
Simplify the right side.
Step 12.2.1
The exact value of is .
Step 12.3
The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from to find the solution in the third quadrant.
Step 12.4
Simplify .
Step 12.4.1
To write as a fraction with a common denominator, multiply by .
Step 12.4.2
Combine fractions.
Step 12.4.2.1
Combine and .
Step 12.4.2.2
Combine the numerators over the common denominator.
Step 12.4.3
Simplify the numerator.
Step 12.4.3.1
Multiply by .
Step 12.4.3.2
Subtract from .
Step 12.5
Find the period of .
Step 12.5.1
The period of the function can be calculated using .
Step 12.5.2
Replace with in the formula for period.
Step 12.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 12.5.4
Divide by .
Step 12.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 13
Step 13.1
The range of cosine is . Since does not fall in this range, there is no solution.
No solution
No solution
Step 14
List all of the solutions.
, for any integer