Precalculus Examples

Find All Complex Solutions tan(theta)=-( square root of 3)/3
Step 1
Multiply each term by a factor of that will equate all the denominators. In this case, all terms need a denominator of .
Step 2
Multiply the expression by a factor of to create the least common denominator (LCD) of .
Step 3
Move to the left of .
Step 4
Simplify .
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Step 4.1
Divide by .
Step 4.2
Multiply by .
Step 5
Take the inverse tangent of both sides of the equation to extract from inside the tangent.
Step 6
Simplify the right side.
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Step 6.1
The exact value of is .
Step 7
The tangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from to find the solution in the third quadrant.
Step 8
Simplify the expression to find the second solution.
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Step 8.1
Add to .
Step 8.2
The resulting angle of is positive and coterminal with .
Step 9
Find the period of .
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Step 9.1
The period of the function can be calculated using .
Step 9.2
Replace with in the formula for period.
Step 9.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 9.4
Divide by .
Step 10
Add to every negative angle to get positive angles.
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Step 10.1
Add to to find the positive angle.
Step 10.2
To write as a fraction with a common denominator, multiply by .
Step 10.3
Combine fractions.
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Step 10.3.1
Combine and .
Step 10.3.2
Combine the numerators over the common denominator.
Step 10.4
Simplify the numerator.
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Step 10.4.1
Move to the left of .
Step 10.4.2
Subtract from .
Step 10.5
List the new angles.
Step 11
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 12
Consolidate the answers.
, for any integer