Math question involving a tricky triangle!?

Question

I have a triangle that has sides of 17.88854382 and 120 and a angle of 105. I need to figure out the other side which appears as the shortest and the other angles. I think there's something wrong in the question, or it's just too hard!

Answer 1

The other side = 125.8219973

The other two angles are:  67.10668487 degrees and 7.893315129 degrees.

To find the first piece, the third side, remember the Pythagorean Theorem:  a^2 + b^2 = c^2. It is actually a simplification because of the right angle. In fact, the broader formula (that works for all triangles) is called the Law of Cosines and says:  a^2 + b^2 - 2ab*cos (angle C) = c^2. (In a right triangle, angle C is 90 degrees and the cosine of it is zero so part disappears.)

So, and I'll assume the angle given is the angle between the two sides given, let's call the side we need to find "c":

c = { a^2 + b^2 - 2ab*cos (angle C) }^0.5....substitute what you know

c = { 17.88854382^2 + 120^2 - 2*17.88854382*120*(cos 105) } ^0.5....do the arithmetic

c = 125.8219973

Good deal. Now to find the other angles.

Just as there is a Law of Cosines, there's a (wait for it...) Law of Sines. It says that the ratio of a side's length to the sine of the angle opposite the side equals the same ratio for the other two pairs of angles and sides. Or, to put it in symbols:

a / sin A = b / sin B = c sin C

You know the side opposite the angle you were given 'cause we just figured it out. So, pick one of the other two sides (let's pick 120 to make the calculator work easy (three button punches vs. 11!)) and see what you have:

120 / sin A = 125.8219973 / sin 105....multiply each side by sin A and by (sin 105) / 125.8219973 to get sin A by itself

sin A = 120 * (sin 105) / 125.8219973....do the arithmetic

angle A = arc sin {120 * (sin 105) / 125.8219973}....do the arithmetic

angle A = 67.10668487 degrees

Now you COULD do the same thing to find the third angle, but you also know the three angles add up to equal 180 degrees and just subtracting is a LOT easier:

A + B + C = 180 degrees....subtract A and C from each side

angle B = 180 - 67.10668487 - 105....do the arithmetic

angle B = 7.893315129 degrees

I wouldn't round the side's length as the problem gave you a long decimal value for one side, but I would round the angles to 67.1 and 7.9 degrees when answering as long as the exact values show in the work.

So, not just too hard, just lengthy and obnoxious! As for the other side appearing to be the shortest, that may just be from them poorly displaying the values you are given. As the other useful Answer states, the third side cannot be shorter than the 17+ side. But if they labelled it poorly and the 17+ refers to the shortest side, everything is cool.