D**u
发帖数: 204 | 1 Inspired by a problem from the math board.
There are 3 vectors in R^3, and the angles between each pair
of them are a, b and c.
Prove this triangle inequality:
sin(a) + sin(b) > sin(c) |
G********d
发帖数: 10250 | 2 无关话题？
【在 D**u 的大作中提到】
: Inspired by a problem from the math board.
: There are 3 vectors in R^3, and the angles between each pair
: of them are a, b and c.
: Prove this triangle inequality:
: sin(a) + sin(b) > sin(c)
|
D**u
发帖数: 204 | 3 无关 for?
【在 G********d 的大作中提到】
: 无关话题？
|
G********d
发帖数: 10250 | 4 本版不讨论数学题吧 只讨论面试题
你自己喜欢做数学题自己做去 不用来显摆
无关 for?
【在 D**u 的大作中提到】
: 无关 for?
|
D**u
发帖数: 204 | 5 This is more like a brainteaser than a research math problem.
Even for 面试题 which this board might favor, it is not
limited to classic probabilty or BS.
【在 G********d 的大作中提到】
: 本版不讨论数学题吧 只讨论面试题
: 你自己喜欢做数学题自己做去 不用来显摆
:
: 无关 for?
|
G********d
发帖数: 10250 | 6 "might" favor
actually means "does not"
【在 D**u 的大作中提到】
: This is more like a brainteaser than a research math problem.
: Even for 面试题 which this board might favor, it is not
: limited to classic probabilty or BS.
|
x******a
发帖数: 6336 | 7 is it true?
if a=pi b=c=pi/2.
【在 D**u 的大作中提到】
: Inspired by a problem from the math board.
: There are 3 vectors in R^3, and the angles between each pair
: of them are a, b and c.
: Prove this triangle inequality:
: sin(a) + sin(b) > sin(c)
|
D**u
发帖数: 204 | 8 I was lazy, should be
sin(a) + sin(b) >= sin(c).
thx for point it out.
【在 x******a 的大作中提到】
: is it true?
: if a=pi b=c=pi/2.
|
L******k
发帖数: 33825 | 9 Law of Sine？？
【在 D**u 的大作中提到】
: Inspired by a problem from the math board.
: There are 3 vectors in R^3, and the angles between each pair
: of them are a, b and c.
: Prove this triangle inequality:
: sin(a) + sin(b) > sin(c)
|
z****i
发帖数: 406 | 10 We have a+b+c = pi.
sin(c) = sin(pi-a-b) = sin(a+b) = sin(a) cos(b) + sin(b)cos(a) <= sin(a) +
sin(b)
【在 D**u 的大作中提到】
: Inspired by a problem from the math board.
: There are 3 vectors in R^3, and the angles between each pair
: of them are a, b and c.
: Prove this triangle inequality:
: sin(a) + sin(b) > sin(c)
|
|
|
D**u
发帖数: 204 | 11 sounds a good suggestion. Then you need to show that
sin(alpha) + sin(beta) >= sin(gamma)
for the dihedral angles.
【在 L******k 的大作中提到】
: Law of Sine？？
|
L******k
发帖数: 33825 | 12 a/sinA=b/sinb=c/sinC=constant d
a=sinA*d
b=sinB*d
c=sinC*d
三角形两边之和大于第三边
vector的话 大于等于第三边
a+b>=c
sinA+sinB>=SinC
我是这么想的 不知道对不
至于那个 constant 是不是有个圆来着 哪个constant=2R
【在 D**u 的大作中提到】
: sounds a good suggestion. Then you need to show that
: sin(alpha) + sin(beta) >= sin(gamma)
: for the dihedral angles.
|
D**u
发帖数: 204 | 13 The law of sin in R^3 is different from the plane case.
In R^3, the law of sin is instead to be:
sin(a)/sin(alpha) = sin(b)/sin(beta) = sin(c)/sin(gamma),
where alpha, beta and gamma are the dihedral angles
between the vectors.
【在 L******k 的大作中提到】
: a/sinA=b/sinb=c/sinC=constant d
: a=sinA*d
: b=sinB*d
: c=sinC*d
: 三角形两边之和大于第三边
: vector的话 大于等于第三边
: a+b>=c
: sinA+sinB>=SinC
: 我是这么想的 不知道对不
: 至于那个 constant 是不是有个圆来着 哪个constant=2R
|
M********t
发帖数: 163 | 14 (操作失误,修改一个TYPO)
不能在这个给你画图.
我把要点叙述一下,你自己画个图:
设三条射线是l1,l2,l3,设交点为O,不失一般性,我们只考虑3个锐角的这一边.我们考虑
l2和l3形成平面@(打不处阿尔法,公理:2直线成1平面)从l1取一点A,不妨设OA=1.
从A向@做一个垂线,垂足为H,AH垂直与@上的任何直线.
在@平面内,从H做l2的垂线,交l3于点C,交l2与点B.
连接AC,BC,AB.
因为AH垂直OB,HB垂直OB,所以平面ABH垂直与OB.
所以角AOB是直角.
那么sinAOB=sin(a)=AB, BC=cos(a)tan(b)=sin(b)*[cos(a)/cos(b)],
sin(c)=sinAOC<=AC.
但是在三角形ABC中, AB+BC>AC.
由于对称性,不妨设[cos(a)/cos(b)]<=1. |
M********t
发帖数: 163 | |
M********t
发帖数: 163 | 16 有个更简单的方法：
设0
那么 a>a' , b>b'.
这是因为 cos(a)
sin(c) = sin(a'+b') |
D**u
发帖数: 204 | 17 Very nice and clear. Better than the solution I have.
' 。
【在 M********t 的大作中提到】
: 有个更简单的方法：
: 设0: 那么 a>a' , b>b'.
: 这是因为 cos(a): sin(c) = sin(a'+b')
|
