Law of sines and triangle solutions

[Neophyte]

Homework Statement

Given triangle ABC with the measure of angel A = 60 degrees, the length of BC = sq rt of 3, and the length of AC = 1/5. How many soulutions are there for the measure of angle B?

The Attempt at a Solution

1. sin(60) / sq rt 3 = sinB / (1/5)

2. sq rt 3 / 2 * 1 / sq rt 3 * 5

3. 5/2 = sin(B)
= Undefined because domain is [-1,1]

It would seem I did something wrong but I keep getting the same thing, any assistance would be great.

 

[danago]

2. sq rt 3 / 2 * 1 / sq rt 3 * 5

Should be:


2. sq rt 3 / 2 * 1 / sq rt 3 * (1/5)

You have just made an arithmetic mistake.

 

[Neophyte]

So if angle B is 5.74 degrees how would you go about finding how many solutions there are? Originally I tried seeing how many times 5.8 went into 90 but I decided that couldn't be right so I started to subtract it from 180 degrees and see if it fit in the domain of arcsin. Which also seems to have provided the wrong answer :(. If the answer were undefined would the number of solutions be zero?

 

[yeongil]

This is the ambiguous case for the Law of Sines -- A S S. ;)

I think you're going about this the wrong way -- I'm not sure you're supposed to find angle B. First, draw a diagram (hint: make AB the base of the triangle). Drop a perpendicular from C to side AB -- this is the height of the triangle (h). How would you go about finding h?


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[Neophyte]

Well assuming that is possible and setting AB as base ( I am not sure how that is assigned the base ) I get..

1. Angle A = 60 Angle B = 5.74 therefore Angle C = 114.26
2. Draw perpendicular and if base is DE now in triangle DEF then Angle F is 57.13
3. 180 - (57.13 + 90 = 147.13) = 32.87 Therefore Angle E is 32.87
4. sin(32.87)= X / sq rt 3 Thus X(h) = 0.94
5.Then from that use a2+b2=c2 to get 1/2 base which ended up being 0.92 so base = 1.84
6.AB = 1.84

Well the problem is I do not know how to find the number of solutions from this either ;/.

The answer is either 0,1,2,3, or can not be determined as I as I already missed it with none of the above. =/ It would seem about 60% of time the answer is two but its not very helpful for when its not lol

 

[yeongil]

I'm not sure what you did ^ up there. Can you draw a diagram and post it here?

I don't think you got the height of the triangle (h as I defined it). Hint: the way to find h doesn't involve the Law of Sines at all.


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