Question

How do you solve `secxtanx=2/3`?

Answer 1

`pi/6, (5pi)/6`

Explanation:

`sec xtan x = 2/3`
`(1/cos x)(sin x/(cos x)) = 2/3`
Cross multiply:
`3sin x = 2 cos^2 x = 2(1 - sin^2 x) = 2 - 2sin^2 x`
Solve this quadratic equation in sin x:
`2sin^2 x + 3sin x - 2 = 0`.
Use the improved quadratic formula (Socratic Search)
`D = d^2 = b^2 - 4ac = 9 + 16 = 25` --> `d = +- 5`
There are 2 real roots:
`sin x = -b/(2a) +- d/(2a) = -3/4 +- 5/4`
`sin x = - 8/4 = - 2` (rejected because < - 1) and
`sin x = 2/4 = 1/2`
Trig table and unit circle -->
`sin x = 1/2` --> arc `x = pi/6` and arc `x = (5pi)/6`
For general answers, add `(k2pi)`