What are the values of sinx and cosx for the acute angle x such that tanx = 4/5 ?

We know that the tangent function is a ratio of the opposite
cathetus and adjacent cathetus.

We'll recall that the opposite side
to the acute angle, in the unit circle, is the y component. But y component, in the unit circle,
is the value of the sine function.

We also know that the adjacent
side to the acute angle, in the unit circle, is the x component. But x component, in the unit
circle, is the value of the cosine function.

tan x = sin x/cos
x

But tan x = 4/5

4/5  = sin x/cos
x

We'll apply the fundamental formula of
trigonometry:

(tan x)^2 + 1 = 1/(cos
x)^2

cos x = 1/sqrt((tan x)^2 + 1)

cos
x = 1/sqrt(16/25 + 1)

cos x = +/- sqrt
(25/41)

cos x = +/- 5sqrt41/41

sin x =
+/-sqrt (1 - 25/41)

sin x = +/-sqrt
16/25

sin x = +/- 4/5