Question

Romeo

問下這兩題，答案分別為3/4，37/143

1. 計算sin²41°+sin²19°+sin41°sin19°的值為
2. 已知△ABC中,AB=5,BC=9
1
1
tan A tan B tan C
B
成等差數列,試求 tan²
2
=

Answer 1

✨ 最佳解答 ✨

qn

1.
cos(x+y)cos(x-y)
=[cos(x)cos(y)-sin(x)sin(y)][cos(x)cos(y)+sin(x)sin(y)]
=cos²(x)cos²(y)-sin²(x)sin²(y)
=[1-sin²(x)][1-sin²(y)]-sin²(x)sin²(y)
=[1-sin²(x)-sin²(y)+sin²(x)sin²(y)]-sin²(x)sin²(y)
=1-sin²(x)-sin²(y)
所以 sin²x+sin²y=1-cos(x+y)cos(x-y)

積化和差 sin(x)sin(y)=-[cos(x+y)-cos(x-y)]/2

原式=1-cos60°cos22°-(cos60°-cos22°)/2
=1-½cos22°-¼+½cos22°
=¾

2.
cosB=(1-tan²(B/2))/(1+tan²(B/2)) (萬能公式)
→ 先算cosB
2cosB/sinB=cosA/sinA + cosC/sinC
2cosB/b=cosA/a+cosC/c
2ac cosB=bc cosA + ab cosC
90cosB=5b cosA + 9b cosC
c=b cosA + a cosB
→b cosA=5-9cosB
a=b cosC + c cosB
→b cosC=9-5cosB
90cosB=5(5-9cosB)+9(9-5cosB)
=106-90cosB
cosB=106/180=53/90
(1-tan²(B/2))/(1+tan²(B/2))=53/90
90-90tan²(B/2)=53+53tan²(B/2)
tan²(B/2)=37/143

Romeo

謝謝！