Question

How do you solve `2tan^2(t)+ 8tan(t)+ 4 = 0`?

Answer 1

`106^@34 + k180`
`149^@50 + k180^@`

Explanation:

Call tan t = x and solve the below quadratic equation for x:
`2x^2 + 8x + 4 = 0`
Use the improved quadratic formula (Socratic Search):
`D = d^2 = b^2 - 4ac - 64 - 32 = 32` --> `d = +- 4sqrt2`
There are 2 real roots:
`tan t = x = -b/(2a) +- d/(2a) = -8/4 +- (4sqrt2)/4 = -2 +- sqrt2`

a. `tan t = -2 + sqrt2 = - 2 + 1.41 = - 0.59`
Calculator -->
`t = - 30^@54`, or `x = 149^@50` (co-terminal)
b. `tan t = - 2 - sqrt2 = - 3.41`.
Calculator gives:
`t = - 73^@66` or `t = 106^@34` (co-terminal)
General answers:
`106^@34 + k180^@`
`149^@50 + k180^@`