Here we are given three sides, 57 mm, 121 mm and 84 mm. If the triangle is not already drawn for you draw it and label the vertices A, B and C, with A opposite the longest side.

We know that if there is an obtuse angle it will face the longest side, so we find that first, so that we can be certain that when we use the Sine Rule to find the other two angles both of them must be acute.

Your calculator says , or -0.452798663324 (with a dot over the 4) when you use the SD key. This is stored in ANS and you do not need to write it down. This is just a division, not an irrational number, and is a recurring decimal with more than twelve digits in the sequence, hence, without scrolling, you can actually see only one dot, over the first number in the sequence.

The negative sign means that A is obtuse, but your calculator takes care of this. So A = cos^{-1} (ANS) Your calculator says 116.92338656 and this is stored in ANS but you will write down 117°. But you will need it again so also store it in one of the variable memories, for example to store it in variable memory D enter ANS *shift* STO D - using memories is explained more fully on the Memories Page.

As here A is obtuse we know that both B and C must be acute and it does not matter which we find next. In fact it does not matter whether A is obtuse or acute provided we use the Cosine Rule to find the largest angle before we use the Sine Rule to find the other two angles, because only the largest angle can be obtuse.

Your calculator says 38.24102038 and this is stored in ANS but you will write down 39°

Your calculator says 0.42001559395 but you need not write it down. So C = sin^{-1} ANS. Your calculator says 24.83559382 but you will write down 25°

Do a quick check: 39 + 25 + 117 = 180

You could in fact have found C just by subtracting A and B from 180 rather than by using the Sine Rule, but doing it this way is almost as quick and does allow you to check your results. Allowing for rounding errors because of the use of irrational numbers you are allowed to be 1° out on this check.

So A = 117°, B = 39° and C = 25°

© Barry Gray April 2016