# unsolvable cosine law question.

• February 2nd 2010, 10:14 PM
tea123456
unsolvable cosine law question.
So i have a triangle with sides 10 and 40, and the angle between them is 70 degrees. I don't know any other measurements of this triangle.

I use cosine law to find the 3rd side, which turns out to be 37.8.

Then I found the other 2 angles with sine law, and I obtain 14 degrees and 84 degrees.

However, the weird thing is the three angles do not add up to 180!!

HELP
• February 2nd 2010, 10:36 PM
Hello tea123456

Welcome to Math Help Forum!
Quote:

Originally Posted by tea123456
So i have a triangle with sides 10 and 40, and the angle between them is 70 degrees. I don't know any other measurements of this triangle.

I use cosine law to find the 3rd side, which turns out to be 37.8.

Then I found the other 2 angles with sine law, and I obtain 14 degrees and 84 degrees.

However, the weird thing is the three angles do not add up to 180!!

HELP

The largest angle has a sine equal to $0.9952$. But there are two possible angles here: one is $84.4^o$, and the other is $180^o-84.4^o$. Try the second one instead.

• February 2nd 2010, 10:39 PM
tea123456
can i ask why there are 2 angles that results the same sine value?

and how come for 14 degrees, there isnt another angle of 180-14?
• February 3rd 2010, 12:34 AM
Hello tea123456
Quote:

Originally Posted by tea123456
can i ask why there are 2 angles that results the same sine value?

I don't know how far your study of trigonometry has gone, but if you're using the Sine and Cosine Rules, you need to know how to handle obtuse angles. The rules are:
$\sin(180^o - \theta)=\sin \theta$
and
$\cos(180^o - \theta)=-\cos \theta$
To understand the reasons for these rules, you'll need to look at the definitions of sine and cosine for angles of any size. These are based on the unit circle. You may find this page helpful.

Quote:

and how come for 14 degrees, there isnt another angle of 180-14?
Because, in a triangle, the smallest angle is always less than $90^o$ (obviously!) and it is always opposite the shortest side (because of the Sine Rule, where each side is in proportion to the sine of the opposite angle).

So the $14^o$ angle is the smallest angle here, because it's opposite the shortest side. And so it's $14^o$, not $166^o$.