Trigonometry: Find solution

Question

If 0 < x < 2π, find the sum of all solutions to the equation 2cos^2(x)-7 cos(x)-4=0.

Show how to do it on a ti-84 if possible.

Answer 1

let y = cosx

then

2y^2 -7y -4 = 0

(2y+1)(y-4)=0

set each factor =0

2y+1=0

y = -1/2

y-4=0

y=4

cosx=-1/2

x = cos^-1(-1/2) = 180+/-60 = 120 or 240 degrees

x = 120+ 360n degrees where n= any integer

or

x = 240+360n degrees where n= any integer

cosx=4 is impossible, with no solutions, as cosines can never be > 1

use any calculator with an inverse cosine function

or

just recognize that's it's a common 30-60-90 degree right triangle

where the reference angle = 60 degrees. Add & subtract 60 from 180 to get the cosine to be negative