Why I can't use acute trigonometric function in obtuse angles?

Question

I need help at understanding why acute trigonometric functions only work with right triangles. I would appreciate any help.

Answer 1

You can use the trigonometric functions to find the angles and side lengths of non-right triangles. The modifications that are required are essentially the law of sines and the law of cosines. These still apply to right triangles, but they take on a simpler form (one of the sines in the law of sines becomes $1$ and the cosine in the law of cosines becomes $0$).

There are two special cases where you can be given three pieces of information and not necessarily be able to determine the triangle up to congruence. The first is being given all three angles and no sides; in this case you know the shape of the triangle but not its size. This is called AAA.

The other is knowing two sides and one of the angles not between those two sides. In this case, if you put point A at the origin, then you know exactly where point B is, but you only know how far point C is from point B, not necessarily where it is. As a result, depending on the exact numbers it can happen that there are two such triangles, one acute and one obtuse. This is called SSA.

Any other combination of three pieces of information can tell you all three sides and all three angles by using some combination of the law of sines and the law of cosines.

Answer 2

The trigonometric functions are defined in terms of ratios of sides of a right angled triangle. $\sin\theta$ is defined to be the ratio between the side opposite an angle $\theta$, and the hypotenuse of a right angled triangle. These ratios change when you change the right angle to an obtuse angle, which is why they no longer hold.