# Law of Cosines oddities

## Homework Statement

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

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/ [Broken]
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. Last edited by a moderator:

AlephZero
Homework Helper
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.

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

AlephZero