why sin x/cos x=tan x(10分)

Let ABC be right-angled triangle,
angle ABC is right-angle,
AC is hypothenuse(斜邊),
angle ACB is x,
AB is opposite side(對邊) of angle x and BC is adjacent side(鄰邊) of angle x.

From definition,
sin x = AB / AC,
cos x = BC / AC and
tan x = AB / BC
but, AB / BC
= (AB / AC) / (BC / AC)
= sin x / cos x
therefore
tan x = AB / BC = sin x / cos x

Cos x =鄰邊(adjacent side) / 斜邊(hypotenuse)

Sin x =對邊(opposite side) / 斜邊(hypotenuse)

Tan x =對邊(opposite side) / 鄰邊(adjacent side)

Let 鄰邊(adjacent side) =a 斜邊(hypotenuse)=b 對邊(opposite side)=c

sin x/cos x = (c/b)/(a/b)
sin x/cos x =(c/b)x(b/a)
sin x/cos x =c/a

tan x=c/a

So sin x/cos x=tan x