中三數學(area, sin cos tan)

Please refer to the solution below:

圖片參考：http://hk.geocities.com/stevieg_1023/AAMM1.gif


圖片參考：http://hk.geocities.com/stevieg_1023/AAMM2.gif

Since questions 1 & 2 are tackled well in the previous work, I am solving Q3.

Q3:
Find, in terms of x and y, sinθ and cosθ if
a)tanθ = y
b)tanθ = x/y

Method:
a)tanθ = y
b)tanθ = x/y [ y <> 0 ]

Let t denotes tanθ, s denotes sinθ and c denotes cosθ,
then t = s/c and s^2 + c^2 =1

From b):
t = x/y ... (i)
From a):
t = y ... (ii)

Multiplying both sides of (ii) to (i) respectively
x = t^2
x = s^2/c^2 [ c <> 0 ]
x(c^2) = s^2
x(c^2) = 1 - c^2
c^2 + x(c^2) = 1
c^2 = 1/(x + 1) [ x <> (-1) ] or c^2 = 1/(y^2 + 1)
c = + or - {(x + 1)^(1/2)/(x + 1)} [ x > (-1) ]

And s^2 = 1 - c^2
So, s^2 = 1 - 1/(x + 1)
s^2 = x/(x + 1) or y^2/( y^2 + 1)
s = + or - { y[(x + 1)^(1/2)/(x + 1)]}

Therefore, sinθ = + or - { y[ (x + 1)^(1/2)/(x + 1) ] }
and, cosθ = + or - { (x + 1)^(1/2)/(x + 1) }