中四 MATHS 20點 (急~!!!)

1.
正弦定律：
a / sinA = b / sinB
Hence, b = a sinB / sinA ...... [1]

正弦定律：
a / sinA = c / sinC
Hence, c = a sinC / sinA ...... [2]

A + B + C = 180°
180° - A = B + C
sin(180° - A) = sin(B + C)
sinA = sin(B + C) ...... [3]

Area = (1/2) b c sinA ...... [4]

把 [1] 和 [2] 代入[4] 中：
Area = (1/2) (a sinB / sinA) (a sinC / sinA) sinA
Area = a² sinB sinC / (2 sinA) ...... [5]

把 [3] 代入 [5] 中：
Area = a² sinB sinC / [2 sin(B + C)]


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2.
a)
a = (20 + 10) cm = 30 cm
b = (20 + 5) cm = 25 cm
c = (10 + 5) cm = 15 cm
s = (30 + 25 + 15)/2 cm = 35 cm

ΔABC面積
= √[s(s - a)(s-b)(s-c)]
= √[35(35 - 30)(35 - 25)(35 - 15)] cm²
= 50√14 cm²
≈ 187.08 cm²

b)
cosB = (a² + c² - b²) / (2ac) (餘弦定律)
cosB = (30² + 15² - 25²) / (2 * 30 * 15)
∠ABC = B = 56.25°

cosA = (b² + c² - a²) / (2bc) (餘弦定律)
cosA = (25² + 15² - 30²) / (2 * 25 * 15)
∠BAC = A = 93.82°

∠ACB = 180° - (56.25° + 93.82°)
∠ACB = 29.93°

c)
紅色部份面積
= [50(√14) - π(5)²(93.82/360) - π(10)²(56.25/360) - π(20)²(29.93/360)] cm²
= 13.05 cm²

http://i1152.photobucket.com/albums/p489/minglamchoi/trig3.jpg
圖片參考：http://i1152.photobucket.com/albums/p489/minglamchoi/trig3.jpg