find all solutions

Question

Find all solutions exactly on the interval [0,2⁢π).

2tan^2 (t)= 3sec(t)

Answer 1

2tan^2(t) = 3sect

tan^2(t) =1- sec^2(t)

2(1-sec^2(t) = 3sect

2 - 2sec^2(t) =3sect

let x = sect

2-2x^2 = 3x

2x^2 -3x -2 = 0

(2x+1)(x-2)= 0

x=-2 = sect = 1/cost

cost = 1/2

t =arccos(1/2) = 60 degrees or 300 degrees

=pi/3 or 5pi/3

check the answer

2tan^2(60) = 6

3sec60= 6 it works

Answer 2

2(sec²t - 1) = 3sect; 2sec²t - 3sect - 2 = 0; sect = 2 or sect = - 1/2(no solutions)

cost = 1/2; t = π/3 or t = 5π/3