F.4 AM

1) A:B:C = 1:2:3
A = 180 x 1 / 6
= 30度
B = 60度
C = 90度

a/sin A : b/sin B : c/sin C
a/0.5 : b/(√3/2) : c
a : √3 b : 2c

a:b:c = 1:√3: 2

2)C = 105度
a/sin A = b/ sin45
√2 a = b

c/sin C = a/sin A
c/ 0.965926 = a/0.5
c = 1.931852 a

a+b+c = 3+√2 +√3
a + √2 a + 1.931852 a = 3 + √2 +√3
4.34607 a = 6.14626
a = 1.414

b = √2 a
b = 2.000

c = 1.931852 a
c = 2.732

2007-12-31 09:24:33 補充：
I think i have made a mistake on question 1.a/sin A = b/sin Ba/0.5 = b/(√3/2)√3 a = ba/b = 1/√3a:b = 1 : √3a/sin A = c/sin Ca/0.5 = ca = 0.5c2a = c a/c = 1/2a:c = 1:2a:b:c = 1:√3: 2

1.sinA/a=sinA/b=sinC/c=k
∴sinA=ak, sinB=bk, sinC=ck
∴a:b:c=sinA:sinB:sinC
=sin30°:sin60°:sin90°
=1:sqrt(3):2

2.C=180°-30°-45°=105°(angle sum of triangle)
sin30°/a=sin45°/b
∴a:b=sin30°:sin45°
=1:sqrt(2)------------(1)
sin30°/a=sin105°/c
∴a:c=sin30°:sin105°
=2:[sqrt(6)+sqrt(2)]--------------(2)
Explanation on (2)
sin105°=sin75°=[sqrt(6)+sqrt(2)]/4,
which can be froun by using compound angle formula sin75°=sin(30°+45°)
By combining (1) and (2)
a:b:c=2:2sqrt(2):[sqrt(6)+sqrt(2)]
∴Let a=2x, b=2sqrt(2)x, c=[sqrt(6)+sqrt(2)]x
∴[2+2sqrt(2)+sqrt(6)+sqrt(2)]x=3+sqrt(2)+sqrt(3)
x=1/sqrt(2)
∴a=sqrt(2), b=2, c=1+sqrt(3)

1)A=k B=2k C=3k
A+B+C=180° (∠ sum of △)
k+2k+3k=180°
6k=180
k=30°
∴A=30° B=60° C=90°
a:b = cot B = cot 60° = √3:3
b:c = sin B = sin 60° = √3/2 = 3:2√3
a:b:c = √3:3:2√3

2)C = 180°-45°-30° = 105°
b = a sin B/sin A = a sin 45°/sin 30° = √2a
c = a sin C/sin A = a sin 105°/sin 30°
a+b+c = a+√2a+a sin 105°/sin 30°
a+b+c=3+√2+√3
a+√2a+a sin 105°/sin 30° = 3+√2+√3
a = √2 = 1.414 (cor. to 4 sig. fig.)
b = √2*√2 = 2
c = √2 sin 105°/sin 30° = 2.732 (cor. to 4 sig. fig.)

P.S.度: Alt+0176 →°