if you let x = tan(theta), then the equation becomes:
3x^3 + 3x^2 - x - 1
this is a cubic equation that can be factored.
i won't get into how i factored it, but will do so if you ask.
the end result of the factoring, however, is:
x can be any of the following:
x = -1
x = 1/sqrt(3)
x = -1/sqrt(3)
i will work this in degrees and then show the final answer in radians as this will be easier for you to see and should make the calculations easier because the angles are common angles for the tan function as you will see.
i could do it in radians but the complexity of understanding radians plus the complexity of showing these common angles in radians takes away from the comprehension of what is happening.
tan(theta) = -1 if and only theta = -45 degrees.
tangent of the angle is negative in the fourth quadrant and the second quadrant.
it's usually best to to work with the positive angle, so add 360 to this until it becomes positive.
-45 + 360 = 315 degrees.
that's the equivalent positive angle in the fourth quadrant.
the reference angle is the equivalent angle in the first quadrant.
to find the reference angle subtract 315 from 360 to get 45 degrees.
the reference angle in the first quadrant is equal to 45 degrees.
the equivalent angle in the second quadrant is equal to 180 - 45 = 135 degrees.
your angles between 0 and 360 degrees that have a tangent of -1 are:
135 degrees and 315 degrees.
135 degrees is the angle in the second quadrant.
315 degrees is the angle in the fourth quadrant.
tan(theta) = 1/sqrt(3) if and only if theta = 30 degrees.
tangent of the angle is positive in the first quadrant and the third quadrant.
30 degrees is already in the first quadrant.
the equivalent angle in the third quadrant is 180 + 30 = 210 degrees.
your angles between 0 and 360 degrees that have a tangent of 1/sqrt(3) are:
30 degrees and 210 degrees.
30 degrees is the angle in the first quadrant.
210 degrees is the angle in the third quadrant.
tan(theta) = -1/sqrt(3) if and only if theta = -30 degrees.
transform this to a positive angle by adding 360 degrees to it to get the equivalent positive angle of 330 degrees.
tangent of the angle is negative in the second quadrant and the fourth quadrant.
this angle is already in the fourth quadrant.
find the reference angle which is the equivalent angle in quadrant 1.
360 - 330 = 30 degrees.
that's the reference angle in quadrant 1.
the equivalent angle in quadrant 2 is equal to 180 - 30 = 150 degrees.
your angles between 0 and 360 degrees that have a tangent of -1/sqrt(3) are:
150 degrees and 330 degrees.
150 degrees is the angle in the second quadrant
330 degrees is the angle in the fourth quadrant
note that 1/sqrt(3) is the same as sqrt(3)/3 after it is simplified by rationalizing the denominator.
you can work with it either way.
i chose to work with it as is without simplifying it.
there is no penalty to doing that as far as i can see.
you'll get the same answer either way.
you now have all the angles between 0 and 360 degrees that satisfy the equation.
those angles are:
for tan(theta) = -1:
135
315
for tan(theta) = 1/sqrt(3)
30
210
for tan(theta) = -1/sqrt(3)
150
330
those same angles sorted in ascending order are:
30
135
150
210
315
330
convert these angles to radians by multiplying them by pi and dividing them by 180 to get:
30 degrees = pi/6 radians
135 degrees = 3pi/4 radians
150 degrees = 5pi/6 radians
210 degrees = 7pi/6 radians
315 degrees = 7pi/4 radians
330 degrees = 11pi/6 radians
the angle shown in radians that satisfy the equation that are in the interval from 0 to 2pi are:
pi/6
3pi/4
5pi/6
7pi/6
7pi/4
11pi/6
those are your solutions.
those are the angles that are shown in the attached graph.
