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第 18 週 · 三角學(一) · 第 6 節

餘弦定理與希羅公式(+本週總關卡)

正弦定理需要「邊對角」配對;只知三邊或兩邊夾角時,改用餘弦定理。它是勾股定理的推廣。再學希羅公式由三邊求面積,最後挑戰 W18 全週 BOSS。配滿三角形圖例。

▶ 第一關: 餘弦定理求第三邊(SAS)
Cbac (find)CBAc² = a² + b² - 2ab cosC
C = 90b = 4a = 3c = ?(example 1)c² = 9 + 16 - 0 = 25, c = 5
120b = 8a = 7c = 13obtuse: -2ab cosC adds (minus a negative)
▶ 第二關: 餘弦定理求角(SSS)
C = ?baccosC = (a² + b² - c²) / (2ab)
b = 7a = 5c = 8cosC = (25 + 49 - 64) / 70= 10 / 70 = 1/7
▶ 第三關: 勾股推廣與希羅公式
C = 90baccosC = 0c² = a² + b²(Pythagoras, W4)
What is given?side + its opp angleSAS or SSSSine LawCosine Law
b = 4a = 5c = 3s = (a+b+c)/2 = 6Area = sqrt(6.3.2.1) = sqrt36 = 6
⚔ BOSS: W18 全週總關卡
W18 Trigonometry ID1 ASTCany angleD2 simplifyidentitiesD3 graphsperiodD4 solveequationsD5-D6 Sine / Cosine Lawsolve any triangle + arearight triangle -> any triangle
BOSS covers 5 sub-topicsgraph periodsin x -> 360solve eqnsinx=1/2 -> 30area1/2 ab sinCASTC quadrant(tip + cos120)simplifycos120=-1/2each subtopic appears in section 4