Since it is a homogeneous equation in sin x and cos x,
we'll divide the equation by (cos x)^2.
(sin x/cos x)^2 -
sinx/cosx - 2 = 0
But the ratio sin x/cos x = tan
x
We'll substitute the ratio by the function tan
x:
(tan x)^2 - tan x - 2 =
0
We'll substitute tan x =
t
t^2 - t - 2 = 0
We'll apply
the quadratic formula:
t1 = [1 +
sqrt(1+8)]/2
t1 = (1+3)/2
t1 =
2
t2 = -1
tan x =
t1
tan x = 2
x =
arctan 2 + k*pi
tan x =
t2
tan x = -1
x = arctan(-1) +
k*pi
x = - arctan 1 + k*pi
x =
-pi/4 + + k*pi
x = 3pi/4 +
k*pi
The solutions of the
equation are: {arctan 2 + k*pi}U{3pi/4 + k*pi}.
No comments:
Post a Comment