SOLUTION: Find all exact solutions on [0, 2π). (Enter your answers as a comma-separated list.) 2 tan2(t) = 3 sec(t)

Algebra ->  Trigonometry-basics -> SOLUTION: Find all exact solutions on [0, 2π). (Enter your answers as a comma-separated list.) 2 tan2(t) = 3 sec(t)       Log On


   

Question 1128735: Find all exact solutions on
[0, 2π).
(Enter your answers as a comma-separated list.)
2 tan2(t) = 3 sec(t)

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

If we restrict the domain of t to [0,2pi) then
t=pi%2F3 or 5pi%2F3

Explanation:
2tan%5E2%28t%29+=+3+sec%28t%29+
2sin%5E2%28t%29%2Fcos%5E2%28t%29+=+3%281%2Fcos%28t%29+%29
2sin%5E2%28t%29+=+3+%28cos%5E2%28t%29%2Fcos%28t%29+%29
2sin%5E2%28t%29+=+3+cos%28t%29+
2%281-cos%5E2%28t%29%29+=+3+cos%28t%29

2+-2cos%5E2%28t%29+=+3+cos%28t%29

2cos%5E2%28t%29+%2B3+cos%28t%29+-2=0 .......factor
2cos%5E2%28t%29+%2B4+cos%28t%29-cos%28t%29+-2=0
%282cos%5E2%28t%29+%2B4+cos%28t%29%29-%28cos%28t%29+%2B2%29=0
2cos%28t%29%28cos%28t%29+%2B2%29-%28cos%28t%29+%2B2%29=0
%282cos%28t%29-1%29%28cos%28t%29+%2B2%29=0
solutions:
case 1:
if 2cos%28t%29-1=0 ->+cos%28t%29=1%2F2 => t=pi%2F3 or t=5pi%2F3 assuming [0,2pi)
case 2:
if cos%28t%29+%2B2%29=0->cos%28t%29+=-2 BUT, cos%28t%29 is element of [-1,1] for all+t; therefore, this case is extraneous and we will disregard it