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Trigonometry Examples
Step 1
Replace the with based on the identity.
Step 2
Reorder the polynomial.
Step 3
Step 3.1
Simplify .
Step 3.1.1
Simplify the expression.
Step 3.1.1.1
Move .
Step 3.1.1.2
Reorder and .
Step 3.1.2
Apply pythagorean identity.
Step 3.1.3
Simplify each term.
Step 3.1.3.1
Rewrite in terms of sines and cosines.
Step 3.1.3.2
Apply the product rule to .
Step 3.1.4
Convert from to .
Step 4
Replace the with based on the identity.
Step 5
Reorder the polynomial.
Step 6
Step 6.1
Simplify the expression.
Step 6.1.1
Move .
Step 6.1.2
Reorder and .
Step 6.2
Apply pythagorean identity.
Step 6.3
Simplify each term.
Step 6.3.1
Rewrite in terms of sines and cosines.
Step 6.3.2
Apply the product rule to .
Step 6.4
Convert from to .
Step 7
Step 7.1
Subtract from both sides of the equation.
Step 7.2
Reorder and .
Step 7.3
Factor out of .
Step 7.4
Factor out of .
Step 7.5
Factor out of .
Step 7.6
Apply pythagorean identity.
Step 7.7
Multiply by .
Step 7.8
Reorder and .
Step 7.9
Rewrite as .
Step 7.10
Factor out of .
Step 7.11
Factor out of .
Step 7.12
Rewrite as .
Step 7.13
Apply pythagorean identity.
Step 8
Step 8.1
Divide each term in by .
Step 8.2
Simplify the left side.
Step 8.2.1
Dividing two negative values results in a positive value.
Step 8.2.2
Divide by .
Step 8.3
Simplify the right side.
Step 8.3.1
Divide by .
Step 9
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 10
Step 10.1
Rewrite as .
Step 10.2
Pull terms out from under the radical, assuming positive real numbers.
Step 10.3
Plus or minus is .
Step 11
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 12
Step 12.1
The exact value of is .
Step 13
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 14
Step 14.1
To write as a fraction with a common denominator, multiply by .
Step 14.2
Combine fractions.
Step 14.2.1
Combine and .
Step 14.2.2
Combine the numerators over the common denominator.
Step 14.3
Simplify the numerator.
Step 14.3.1
Multiply by .
Step 14.3.2
Subtract from .
Step 15
Step 15.1
The period of the function can be calculated using .
Step 15.2
Replace with in the formula for period.
Step 15.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 15.4
Divide by .
Step 16
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 17
Consolidate the answers.
, for any integer