Question

已知△ABC的三个内角A，B，C所对的边分别为a，b，c，√3b=2asinB，且向量AB•向量AC＞0. （1）求

已知△ABC的三个内角A，B，C所对的边分别为a，b，c，√3b=2asinB，且向量AB•向量AC＞0.
（1）求∠A的度数
（2）若cos(A-C)+cosB=√3/2，a=6，求△ABC的面积.

Answer 1

解：
(1)
∵a/sinA=b/sinB=2R
∴a=2RsinA，b=2RsinB
∵(√3)b=2asinB
∴(√3)2RsinB=2×2RsinA×sinB
∴(√3)sinB=2sinAsinB
∵在△ABC中，0<∠B<180°
∴sinB≠0
∴(√3)=2sinA
∴sinA=(√3)/2.
∵向量AB•向量AC>0
∴|向量AB|×|向量AC|×cosA>0
∵|向量AB|>0，|向量AC|>0
∴cosA>0
∴∠A=60°.
(2)
∵在△ABC中，A+B+C=π
∴B=π-(A+C)
∵cos(A-C)+cosB=(√3)/2
∴cos(A-C)+cos[π-(A+C)]=(√3)/2
∴cos(A-C)-cos(A+C)=(√3)/2
∴(cosAcosC+sinAsinC)-(cosAcosC-sinAsinC)=(√3)/2
∴2sinAsinC=(√3)/2
∵∠A=60°
∴sinA=(√3)/2
∴2[(√3)/2]sinC=(√3)/2
∴2sinC=1
∴sinC=1/2.
∵∠A=60°
∴0°<∠C<120°
∴∠C=30°
∴∠B=180°-∠A-∠C=90°
∴△ABC是以∠B为直角的直角三角形，sinB=1
∵(√3)b=2asinB，a=6
∴(√3)b=12
∴b=4√3.
∵△ABC是以∠B为直角的直角三角形
∴a^2+c^2=b^2
∴36+c^2=48
∴c^2=12
∴c=2√3.
∴S△ABC=(1/2)ac=6√3.

Answer 2

(1)
√3b=2asinB
sinB = √3b/(2a)
By sine-rule
a/sinA = b/sinB
a/sinA = b/(√3b/(2a))
[√3b/(2a)].a = bsinA
sinA =√3/2
A = π/3 ( AB.AC >0 )
(2)
cos(A-C)+cosB= √3/2
cos(A-(π-A-B)+ cosB = √3/2
cos(B-π/3)+ cosB = √3/2
(1/2)cosB+ (√3/2)sinB + cosB = √3/2
(√3/2)sinB-(1/2)cosB = √3/2
sin(B-π/6)= sin(π/3)
B = π/2
C = π/6

△ABC的面积
(1/2)a. (atanA)
= (1/2) a. (atanA)
=(1/2).6 (6. tan(π/3))
= 18√3

Answer 3

In current society, people have known the importance of keeping healthy and good life habit. In the past, people ate three meals at home, and they ate more vegetables and less meat and fish. They were very strong and seldom went to see doctors.Now life is better, people eat more meat, fish, eggs and other food with more fat. They often go out for good meals with their families and friends. But they get easy to be ill, Why?I think more meat and fish, less exercise are bad for health. Good rest, less meat and fish, more vegetables and enough exercise are good for health. So I exercise every day. My eating habits are pretty good. I eat a lot of vegetables. I eat fruit and drink milk every day. I drink a cup of water 15 minutes before each meal and eat much vegetables and fruit. Of course, I love junk food, too, and I eat it once a week. And I sleep nine hours every day.The better we get into good eating habits, the happier our life becomes