# Thread: Why does first cosine theorem work for a triangle that doesn't exist?

1. ## Why does first cosine theorem work for a triangle that doesn't exist?

Given $\displaystyle \triangle ABC$, where $\displaystyle b=3$, $\displaystyle c=3$, $\displaystyle B=45^\circ$, and $\displaystyle C=90^\circ$, what's the length of a?

The triangle doesn't exist, because the hypotenuse cannot be the same length as one of the other sides. But when you put it in the first cosine theorem, it gives an answer

$\displaystyle a=3\cos 90^\circ + 3\cos 45^\circ=\frac{3\sqrt{2}}{2}$

???

2. Originally Posted by chengbin
Given $\displaystyle \triangle ABC$, where $\displaystyle b=3$, $\displaystyle c=3$, $\displaystyle B=45^\circ$, and $\displaystyle C=90^\circ$, what's the length of a?

The triangle doesn't exist, because the hypotenuse cannot be the same length as one of the other sides. But when you put it in the first cosine theorem, it gives an answer

$\displaystyle a=3\cos 90^\circ + 3\cos 45^\circ=\frac{3\sqrt{2}}{2}$

???
You should know that, for a right angled triangle, the law of cosines reduces to $\displaystyle c^{2} = a^{2}+b^{2}$

This will give you $\displaystyle a = 0$

3. what is the "first cosine theorem" ?

never heard of it.

4. Umm, just realized that this doesn't exist (???)

What I learned was

$\displaystyle a=b\cos C + c\cos B$
$\displaystyle b=a\cos C + c\cos A$
$\displaystyle c=a\cos B + b\cos A$