sin and cos in the triangle

Question

I have a question and wanted to discuss it.If in a triangle you know dhe sin and cos is it possible to find the length of the sides?If yes,how?I know that sin and cos are the raport of the sides and hypotenuse but we only know the raport not the actual length of the side,right? thank you

Answer 1

You cannot determine the length of the sides because, as you said, you only know the ratio. The sin and cos of one angle of a triangle gives you the shape up to an ambiguity of scale. It's like you took a picture of the triangle, but you don't know how far away you were when you took the picture.

Answer 2

Knowing the values of sine and cosine, you will only know what the ratio of the respective sides are. If you have no other information such as the length of one of the sides, you cannot deduce the length of the others, as suppose two triangles have opposite and hypotenuse lengths of 1,2 and 2,4 each respectively. Then the sine for both of these triangles will simply be $\frac{1}{2}$, and the cosine $\sqrt{\frac{3}{4}}$ and they will be indistinguishable without knowing anymore information.