Question

How do you find the solution to `tan^2theta-5tantheta+6=0` if `0<=theta<2pi`?

Answer 1

`63^@43. 71^@56', 243^@43, 251^@56`

Explanation:

Solve this quadratic equation for tan t by the improved quadratic formula (Socratic Search):
`tan^2 t - 5tan t + 6 = 0`
`D = d^2 = b^2 - 4ac = 25 - 24 = 1` --> `d = +- 1`
There are 2 real roots:
`tan t = -b/(2a) +- d/(2a) = 5/2 +- 1/2`
`tan t = 6/2 = 3` and `tan t = 4/2 = 2`.
Use calculator and unit circle -->
a. tan t = 3 --> arc `t = 71^@56` and arc `t = 71.56 + 180 = 251^56`
b. tan t = 2 --> arc `t = 63^@43`, and `t = 180 + 63.43 = 243^@43`