中五解三角形

1)
在ΔABC中, b = 21mm, A = 120° 及sinB : sinC = 1 : 2, 解ΔABC

解：
sinB / b = sinC / c (正弦定律)
1 / (21 mm) = 2 / c
c = 42 mm

a² = b² + c² - 2*b*c*cosA (餘弦定律)
a² = 21² + 42² - 2*21*42*cos120° mm²
a² = 3087 mm²
a = 21√7 mm
or a = 55.56 mm

sinA / a = sinB / b (正弦定律)
sin120° / 21√7 = sinB / 21
B = 19.11°

A + B + C = 180° (Δ內角和)
120° + 19.11° + C = 180°
C = 40.89°

A = 120°, B = 19.11°, C = 40.89°
a = 21√7 mm = 55.56 mm, b = 19.11°, c = 40.89°


2)
在ΔABC中, A = 50° 及 B = 60°, 若ΔABC的面積是 82cm², 解ΔABC

解：
A + B + C = 180° (Δ內角和)
50° + 60° + C = 180°
C = 70°

ΔABC面積 = 82 cm²
(1/2)*a*b*sin70° = 82cm²
ab = 164/sin70°...... [1]

同理，(1/2)*b*c*sin50° = 82 cm²
bc = 164/sin50°...... [2]

同理，(1/2)*c*a*sin60° = 82 cm²
ca = 164/sin60°...... [3]

[1]´[2]´[3] :
(abc)² = 164³/sin50°sin60°sin70°
abc = √(164³/sin50°sin60°sin70°) ...... [4]

[4]/[1] :
c = 15.24 cm

[4]/[2] :
a = 12.43 cm

[4]/[3] :
b = 14.05 cm

A = 50°, b = 60°, C = 70°
a = 12.43 cm, b = 14.05 cm, c = 15.24 cm