Trigonometry Maths

a)
Let angle OAC be a
AC² + AO² - 2(AO)(AC)(cos a) = OC²
6² + 3² - 2(6)(3)(cos a) = 4²
-2(6)(3)(cos a) = -29
cos a = (-29) / (-36)
a = 36.336°

b)
i)
tan40° = BC / OC
BC = 4(tan40°)

tan30° = BC / CD
CD = 4(tan40°) / tan30°
CD = 5.813m

ii)
Let angle CAD be A
AC² + AD² - 2(AC)(AD)(cos A) = CD²
6² + 8² - 2(6)(8)(cos A) = 5.813²
A = 46.395°

iii)
from above a) and bii)
Angle EAD = Angle CAD – Angle OAC
= A - a
= 46.395° - 36.336°
= 10.059°

Angle AED = 180° - θ
ED / sin(A-a) = 8 / sin(180° - θ)
(note: sin(180° - θ) = sinθ)
ED = 8(sin10.059°) / sinθ

CE / (sin a) = 6 / sinθ
CE = 6(sin36.336°) / sinθ

CD = CE + ED
CD = [6(sin36.336°) / sinθ] + [8(sin10.059°) / sinθ]
5.813 = [6(sin36.336°) + 8(sin10.059°)] / sinθ
sinθ = [6(sin36.336°) + 8(sin10.059°)] / 5.813
θ = 58.425°

(a)
Angle OAC=arccos((6^2+3^2-4^2)/(2*6*3)) [by cosine rule]
Angle OAC=36.33deg

(b)
(i)
BC=4*tan(40deg)=3.356m
CD=BC/tan(30deg)=5.813m
(ii)
Angle CAD=arccos((6^2+8^2-CD^2)/(2*6*8)) [by cosine rule]
Angle CAD=46.4deg

Angle ACD=arccos((6^2+CD^2-8^2)/(2*6*CD))=85.234deg
Angle CEA=Theta=180deg-36.33deg-85.234deg=58.436deg