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Trigonometry Examples
Step 1
Step 1.1
Divide each term in by .
Step 1.2
Simplify the left side.
Step 1.2.1
Cancel the common factor of .
Step 1.2.1.1
Cancel the common factor.
Step 1.2.1.2
Divide by .
Step 2
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 3
Step 3.1
The exact value of is .
Step 4
Multiply both sides of the equation by .
Step 5
Step 5.1
Simplify the left side.
Step 5.1.1
Cancel the common factor of .
Step 5.1.1.1
Cancel the common factor.
Step 5.1.1.2
Rewrite the expression.
Step 5.2
Simplify the right side.
Step 5.2.1
Combine and .
Step 6
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 7
Step 7.1
Multiply both sides of the equation by .
Step 7.2
Simplify both sides of the equation.
Step 7.2.1
Simplify the left side.
Step 7.2.1.1
Cancel the common factor of .
Step 7.2.1.1.1
Cancel the common factor.
Step 7.2.1.1.2
Rewrite the expression.
Step 7.2.2
Simplify the right side.
Step 7.2.2.1
Simplify .
Step 7.2.2.1.1
To write as a fraction with a common denominator, multiply by .
Step 7.2.2.1.2
Combine fractions.
Step 7.2.2.1.2.1
Combine and .
Step 7.2.2.1.2.2
Combine the numerators over the common denominator.
Step 7.2.2.1.3
Simplify the numerator.
Step 7.2.2.1.3.1
Multiply by .
Step 7.2.2.1.3.2
Subtract from .
Step 7.2.2.1.4
Multiply .
Step 7.2.2.1.4.1
Combine and .
Step 7.2.2.1.4.2
Multiply by .
Step 8
Step 8.1
The period of the function can be calculated using .
Step 8.2
Replace with in the formula for period.
Step 8.3
is approximately which is positive so remove the absolute value
Step 8.4
Multiply the numerator by the reciprocal of the denominator.
Step 8.5
Multiply by .
Step 9
The period of the function is so values will repeat every radians in both directions.
, for any integer