Consider a triangle where a=80 degrees, a=2 cm and b=4 cm. how many possible triangles have the above properties

0 You have a triangle with an angle of 80 degrees and the opposite side being 2 cm. You also have an adjacent side that's 4 cm long. Those sides and the angle specified make for an impossible situation. Let's prove that using the law of sines. The law of sines states that in a triangle, the ratio of the sine of an angle to the length of the opposite side is a constant within that triangle. So sin (A) / a = sin (B) / b = sin (C) / c Let's just deal with those sides and angles involving A and B. sin (A) / a = sin (B) / b Now substitute the known values sin (80) / 2 = sin (B) / 4 0.984807753/2 = sin (B) / 4 0.492403877 = sin (B) / 4 4*0.492403877 = 4*sin (B) / 4 1.969615506 = sin (B) And we now have an impossible situation since the sine of an angle has to be in the range - 1 to 1 and the value 1.969615506 is obviously well beyond that range. Therefore there are no triangles with that angle and side lengths.