00:01
So this question asks us that given a triangle where we know all three sides, can we find the angles a, b, and c? and so what we can use for this is both the fact that from pythagorean's theorem, the a squared plus b squared, so the two smaller sides will always equal the third side square.
00:25
And also the cosine theta is equal to the adjacent side of an angle divided by.
00:31
The hypotenuse where the hypotenuse is the largest side and we can combine these in order to get the expression that an angle theta is equal to the inverse cosine of adjacent side squared plus the other adjacent side squared of where the adjacent sides are the adjacent sides to the angle we're interested in minus the opposite side to the angle we're interested in squared all over two times both adjacent sides.
01:15
And so seeing how this would work in this problem, for example, for angle a, as i have defined it, our two adjacent sides are 91 centimeters and 193 centimeters.
01:29
So we would have inverse cosine of 91 squared plus 193 squared, minus and then our opposite side is 154 squared all over 2 times 91 times 193 and so plug it in all of our numbers here you can get that this is just going to equal the inverse cosine of 21 ,814 divided by 2 times 91 times 193.
02:24
And plugging this into a calculator, we can get that a is equal to 51 .69 degrees.
02:43
So now applying this to our next angle, angle b, we can use b equals cosine inverse.
02:52
And looking at which are our adjacent sides now and which is our opposite.
02:57
We can see that we defined angle b such that the adjacent sides are 91 and 154.
03:07
So if we put those in, 91 squared plus 154 squared minus 193 squared.
03:20
All of that is going to be over 2 times our adjacent sides.
03:31
That's going to equal inverse cosine of plugging in our numbers here, negative 52, 2 times over 2 times 91 times 154.
04:11
The inverse cosine of this times 954.
04:25
And putting this into our calculator, we get 100 .8 degrees...