if tgx=1/3, the is the value of sinx or cox
I know, the cotgx is inverted value, so its 3 (i hope). I know, that i can calculete arctg 1/3 and after it calculate the sinx and cosx, but i think, that there is better solution...
The enunciation of the problem has requested the values of sin x and cos x, not the values of the angle x!
x e (MPI; (3*MPI)/2)
We'll apply Pythagorean identity:
(sin x)^2 + (cos x)^2 = 1
We'll divide by (cos x)^2:
(tan x)^2 + 1 = 1/(cos x)^2
We'll replace tan x by 1/3
1/9 + 1 = 1/(cos x)^2
10/9 = 1/(cos x)^2 => cos x = +sqrt 9/10
Since it is not specified what is the range of values of x, the values of cosine function may be positive or negative.
cos x = -3sqrt10/10 or cos x = +3sqrt10/10
sin x = sqrt (1 - 9/10)
sin x = sqrt10/10 or sin x = -sqrt10/10
The values of sin x and cos x, when tan x = 1/3, are: sin x = sqrt10/10 or sin x = -sqrt10/10 and cos x = -3sqrt10/10 or cos x = +3sqrt10/10.