Trigonometry Examples

Solve for a cot(a/3)=( square root of 3)/3
Step 1
Take the inverse cotangent of both sides of the equation to extract from inside the cotangent.
Step 2
Simplify the right side.
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Step 2.1
The exact value of is .
Step 3
Since the expression on each side of the equation has the same denominator, the numerators must be equal.
Step 4
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 5
Solve for .
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Step 5.1
Multiply both sides of the equation by .
Step 5.2
Simplify both sides of the equation.
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Step 5.2.1
Simplify the left side.
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Step 5.2.1.1
Cancel the common factor of .
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Step 5.2.1.1.1
Cancel the common factor.
Step 5.2.1.1.2
Rewrite the expression.
Step 5.2.2
Simplify the right side.
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Step 5.2.2.1
Simplify .
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Step 5.2.2.1.1
To write as a fraction with a common denominator, multiply by .
Step 5.2.2.1.2
Simplify terms.
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Step 5.2.2.1.2.1
Combine and .
Step 5.2.2.1.2.2
Combine the numerators over the common denominator.
Step 5.2.2.1.2.3
Cancel the common factor of .
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Step 5.2.2.1.2.3.1
Cancel the common factor.
Step 5.2.2.1.2.3.2
Rewrite the expression.
Step 5.2.2.1.3
Move to the left of .
Step 5.2.2.1.4
Add and .
Step 6
Find the period of .
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Step 6.1
The period of the function can be calculated using .
Step 6.2
Replace with in the formula for period.
Step 6.3
is approximately which is positive so remove the absolute value
Step 6.4
Multiply the numerator by the reciprocal of the denominator.
Step 6.5
Move to the left of .
Step 7
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 8
Consolidate the answers.
, for any integer