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Trigonometry Examples
Step 1
Replace the with based on the identity.
Step 2
Reorder the polynomial.
Step 3
Substitute for .
Step 4
Step 4.1
Rewrite.
Step 4.2
Simplify by adding zeros.
Step 4.3
Apply the distributive property.
Step 4.4
Multiply by .
Step 5
Subtract from both sides of the equation.
Step 6
Subtract from both sides of the equation.
Step 7
Subtract from .
Step 8
Step 8.1
Factor out of .
Step 8.1.1
Factor out of .
Step 8.1.2
Factor out of .
Step 8.1.3
Rewrite as .
Step 8.1.4
Factor out of .
Step 8.1.5
Factor out of .
Step 8.2
Factor.
Step 8.2.1
Factor using the AC method.
Step 8.2.1.1
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 8.2.1.2
Write the factored form using these integers.
Step 8.2.2
Remove unnecessary parentheses.
Step 9
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 10
Step 10.1
Set equal to .
Step 10.2
Subtract from both sides of the equation.
Step 11
Step 11.1
Set equal to .
Step 11.2
Subtract from both sides of the equation.
Step 12
The final solution is all the values that make true.
Step 13
Substitute for .
Step 14
Set up each of the solutions to solve for .
Step 15
Step 15.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 15.2
Simplify the right side.
Step 15.2.1
The exact value of is .
Step 15.3
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 15.4
Subtract from .
Step 15.5
Find the period of .
Step 15.5.1
The period of the function can be calculated using .
Step 15.5.2
Replace with in the formula for period.
Step 15.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 15.5.4
Divide by .
Step 15.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 16
Step 16.1
The range of cosine is . Since does not fall in this range, there is no solution.
No solution
No solution
Step 17
List all of the solutions.
, for any integer