Solve in x radians for exact solutions over the interval [0,2pi) : cot3x = sqrt 3

Question

cot3x = sqrt3   Over the interval [0,2pi)    -   Ive seen examples done but the sqrt throws me hard.

Answer 1

Cotangent is reciprocal of tangent so cot3x=1/tan3x=1/sqrt(3). Draw a right-angled triangle containing an angle of 30 degrees. If the hypotenuse has length 2 the shortest side opposite the 30 degree angle has length 1 so sine30=1/2. The other side has length sqrt(4-1)=sqrt(3). Tangent of 30 is opposite over adjacent=1/sqrt(3) and cot30=sqrt(3). So if 3x=30, x=10. 30 degrees is 180/6=(pi)/6 radians, so x=(pi)/18. Over the range 0 to 2(pi) (0 to 360 degrees), the tangent has positive values between 0 and (pi)/2 (1st quadrant 0 to 90 deg) and between (pi) and 3(pi)/2 (3rd quadrant 180 and 270 deg), so tan210=1/sqrt(3), cot210=sqrt(3) and 210=7(pi)/6, 3x=7(pi)/6, making x=7(pi)/18 (70 deg). The two answers are x is (pi)/18 (0.1745 rad) and 7(pi)/18 (1.2217 rad) (10 and 70 deg).