Law of Cosines Oddities: Solving Triangles with Given Sides and Angles

[pugfug90]

Homework Statement

Solve triangle having indicated sides and angles.
a=6
b=9
C=45degrees

Homework Equations

Law of Cosines
Law of Sines

The Attempt at a Solution

I did c^2=9^2+6^2-(2*9*6*Cos45 degrees)
c=6.37
...
Then I did 6.37/cos45 =9/SinB.. That proportion would give me 87.5 derees.. However, the back of my book and..
http://www.trig.ionichost.com/
Says that B is 93.3 degrees..

However, if I were to approach the problem by solving for A (smallest angle) instead of immediately to B (mid sized).. I would get 41.7 degrees..

Then I could do B=180-A-C..or 93.3 degrees..

But still, if I set up a ratio of sin41.7degs/6=sinB/9, that's not 93.3, or 87.5, but 86.2

I'd like to know why math is stupid. And next time, what I should do to know whether 93.3, 87.5, or 86.2 would be the right answer.

 

[AlephZero]

Your problem is you are not working accurately enough.

My calculator gives c = 6.37456

If you use c = 6.37 you get sin B = (9 sin 45)/ 6.37 = 0.99905

If you use c = 6.37456 you get sin B = 0.9983373

If you look at the graph of sin x, it is almost "horizontal" when x is near to 90 degrees and sin x is nearly 1.0

if sin B = 0.9983373 then B = 86.70 or 93.30

if sin B = 0.99905 then B = 87.50 or 92.50

The small change in the value of sin B causes a big change in the value of B

When you found the smallest angle of the triangle by the sine rule, you avoided this problem for two reasons:

1 the slope of the graph of sin x is steeper so there is less error in going from sin x to x
2 you know the smallest angle of a triangle must be less than 90 degrees, so the other solution (A = 130.3) is not possible.

 

[pugfug90]

Ok.. So when given SSA, always find the S of the given A.. then find the A of the other smallest S?

 

[AlephZero]

Yes that would work.

A more general message to take away from this is: If you are working with any function and you are close to the point where the graph is horizontal (zero slope), then be careful if you are using the inverse function. For example sin x near 90 degrees, or cos x near 0 degrees.

 

[pugfug90]

Also!
You can only have the possibility of having 2 solutions with SSA, right?