Question

How do you find the solution to `2cot^2theta-13cottheta+6=0` if `0<=theta<2pi`?

Answer 1

`9^@46, 63^@43, 189^@46, 243^@43`

Explanation:

Solve this quadratic equation for cot t
`f(t) = 2cot^2 t - 13cot t + 6 = 0`
`D = d^2 = b^2 - 4ac = 169 - 48 = 121` --> `d = +- 11`
There are 2 real roots:
`cot t = -b/(2a) +- d/(2a) = 13/4 +- 11/4 = (13 +- 11)/4`
`cot = 24/4 = 6` --> `tan t = 1/(cot t) = 1/6`
`cot t = 2/4 = 1/2` --> `tan t = 1/(cot t) = 2`
Use calculator and unit circle -->
a. `tan t = 1/6` --> `t = 9^@46` and `t = 180 + 9.46 = 189^@46`
b. tan t = 2 -->`t = 63^@43` and `t = 180 + 63.43 = 243^@43`

Answers for (0, 360):
`9^@46, 63^@43, 189^@46, 243^@43`