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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 as .
Step 4.2
Check that the middle term is two times the product of the numbers being squared in the first term and third term.
Step 4.3
Rewrite the polynomial.
Step 4.4
Factor using the perfect square trinomial rule , where and .
Step 5
Set the equal to .
Step 6
Add to both sides of the equation.
Step 7
Substitute for .
Step 8
Take the inverse cotangent of both sides of the equation to extract from inside the cotangent.
Step 9
Step 9.1
The exact value of is .
Step 10
The cotangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from to find the solution in the fourth quadrant.
Step 11
Step 11.1
To write as a fraction with a common denominator, multiply by .
Step 11.2
Combine fractions.
Step 11.2.1
Combine and .
Step 11.2.2
Combine the numerators over the common denominator.
Step 11.3
Simplify the numerator.
Step 11.3.1
Move to the left of .
Step 11.3.2
Add and .
Step 12
Step 12.1
The period of the function can be calculated using .
Step 12.2
Replace with in the formula for period.
Step 12.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 12.4
Divide by .
Step 13
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 14
Consolidate the answers.
, for any integer