I have no problem with right angle triangles, but when it's not, I don't know how to use trigonometry. Because there are $2$ adjacent sides to each angle, I don't know which one to use.

The question is: A triangles sides are given by coordinates: $A(0,0), B(-2,4), C(4,5)$. I need to get the angles at each vertex. So I got the length of each side.
$AB$ is: $\sqrt{20}$
BC is: $\sqrt{37}$
AC is: $\sqrt{41}$

I used geogebra for the points, and I got the angles for them, and then I tried sin, cos, tan, none worked, then I tried the cosine function: $c^2=a^2+b^2-2ab\cos(C)$ But I still didn't get the answer.

Could anyone help with solving the question, and explaining in this case what rule to use? Also, how do I know which side is $c$, $a$ and $b$ when the triangle is not right-angled?

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    $\begingroup$ The cosine rule you quoted should work. Choose an angle $C$ that you want to calculate, then make $c$ the side that is opposite this angle, and then $a$ and $b$ can be the other two sides. $\endgroup$ – John Doe Sep 28 '17 at 15:50
  • $\begingroup$ By convention, $c$ is the length of $\overline{AB}$, $b$ is the length of $\overline{AC}$, and $a$ is the length of $\overline{BC}$. $\endgroup$ – N. F. Taussig Sep 28 '17 at 16:00
  • $\begingroup$ Oh, so C could be anything I chose, as long as I use c for the opposite side? @JohnDoe $\endgroup$ – JohnFire Sep 28 '17 at 16:02
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    $\begingroup$ @JohnFire yep, exactly :) $\endgroup$ – John Doe Sep 28 '17 at 17:23

By law of cosines $$\cos\measuredangle A=\frac{20+41-37}{2\sqrt20\sqrt{41}}=\frac{6}{\sqrt{205}}.$$ Thus, $$\measuredangle A=\arccos\frac{6}{\sqrt{205}}=65.22...^{\circ}.$$ $\measuredangle B$ and $\measuredangle C$ we can get by the similar way.

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    $\begingroup$ why do you wrote the equal sign? are you God, that you know the other (infinity) Digits? (joke) $\endgroup$ – Dr. Sonnhard Graubner Sep 28 '17 at 15:56
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    $\begingroup$ Sonnhard, I wrote $...$ :). $\endgroup$ – Michael Rozenberg Sep 28 '17 at 16:01
  • $\begingroup$ Thank you very much. Could I use the law of sine to get the remaining 2 angles, or the law of cosine is better? $\endgroup$ – JohnFire Sep 28 '17 at 16:03
  • $\begingroup$ @JohnFire I think it's better to use the law of cosines because you'll get the measure of the angle immediately, while by using of the law of sines you'll get two values and only one of them we can take. $\endgroup$ – Michael Rozenberg Sep 28 '17 at 16:06

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