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

1 Answer
Oct 21, 2015

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#