Question

How do you find the solution for `cosx - 1/cosx = 5/6` for [-180,180]?

Answer 1

Solve `cos x - 1/(cos x) = 5/6`

Ans: `+- 131^@81`

Explanation:

`(cos^2 x - 1)/cos x = 5/6`
`6cos ^2 x - 6 = 5cos x.`
Call cos x = t and solve the quadratic equation for t.
`y = 6t^2 - 5t - 6 = 0` (1)
Use the new Transformation Method.
Transformed equation --> `y' = t^2 - 5t - 36 = 0.`
Factor pairs of (-36) --> (-3, 12)(-4, 9). This sum is 5 = -b. Then y1 = -4 and y2 = 9. Therefor, the 2 real roots of equation (1) are:
`t1 = (y1)/a = -4/6 = -2/3` and `t2 = (y2)/a = 9/6 = 3/2.`
a. `cos x = t2 = 3/2` Rejected because > 1
b. `cos x = t1 = -2/3` --> `x = +- 131^@81`