find all real numbers in the interval [0,2pi] that satisfy the equation: 3sec^2x tanx= 4 tan x?

Question

round to the nearest tenth

Answer 1

3sec^2x tanx-4 tan x = 0
tanx (3 sec^2 x - 4) = 0

tan x = 0 OR (3 sec^2 x - 4 = 0
The solutions to tan x = 0 are x = 0, pi, 2pi

The second equation simplifies as follows:

sec^2 x = 4/3
cos^2 x = 3/4
cos x = +- sqrt 3 / 2
x = pi/6, 5pi/6, 7pi/6, 11pi/6

Answer 2

3 sec^2 x tan x = 4 tan x
<=> tan x = 0 or 3 sec^2 x = 4
<=> tan x = 0 or cos^2 x = 3/4
<=> tan x = 0 or cos x = ± √3 / 2
<=> x ∈ {0, π, 2π, π/6, 5π/6, 7π/6, 11π/6}.

If you really want ugly decimal approximations, you can calculate them yourself.

Answer 3

3sec(x)^2 * tan(x) = 4tan(x)
3sec(x)^2 = 4
sec(x)^2 = (4/3)
sec(x) = 2sqrt(1/3)
1/cos(x) = 2/sqrt(3)
cos(x) = sqrt(3)/2
x = (pi/6), (11pi/6), (5pi/6), (7pi/6)

Since tan(0, pi, or 2pi) gets you 0

ANS : x = 0, 3.1, 6.3, .5, 5.6, 2.6, 3.7