By definitions of the (co)tangent and cosecant function,
Turn everything into fractions with common denominators:
Recall that , so we can simplify both sides a bit.
On the left:
On the right:
(as long as , which happens in the interval when or )
So we have
What it is, is, a/sin A = b/sin B = c/sin C. This is the law. Basically, at the moment, it says sin
Okay, so, first you should identify the right angle. Obviously, it's < C. This means that they hypotenuse, hyp as they use it in class, is opposite it. Side AB.
It gives us the side c, or CB underneath. c=12. Sine is simple to remember, it's the opposite over the hypotenuse.
So first, we should acknowledge that sin A = sin 75°, sin B = sin 26°, and sin C = sin 90°.
So, here we go. sin 75° is opposite to side c, (12), so we line them up like this.
sin 75° sin 26° sin 90°
12 b a
We're going to find side b and side a. Because we're given side c, we can use it in two ways. As an adjacent side, or an opposite side.
Problem here is the hypotenuse, we don't know it. Now, to use sin, we need side a, which is the hypotenuse, we don't have. So we have to find that side first.
We have the adjacent side, (we're using angle B.) and the angle. We have the adjacent, and need the opposite, so for this we're going to use tan.
tan 26°/1 = 12/x
We cross multiply here.
We need to get x alone, so we divide tan 26° from both sides as so:
tan 26° tan 26°
Dividing tan 26° from 12 will give us 24.60 (Rounded)
So, side b is 24.50.
This helps us a lot now, because we can use that side to find the hypotenuse.
We'll use angle B again, to be consistent, and we'll use sine, opposite/hypotenuse.
Sine 26° 24.50
Just like we did above with tan, we cross multiply, giving us: sin 26°x = 24.50
Divide sin 26° from both sides to get the x alone. sin 26° is .4383711468.
Divide that from 24.50, and we get 55.89 (once again, rounded to the correct number of significant digits.)
So, we have our sides, which was what we needed to figure out. side a is 55.89, and side b is 24.50, and side c is 12.
The Law of Sines corresponds with the Law of Cosines. The key of the Law of Cosines is if you have two angles and a side, you can find out everything else about the triangle.
And that's what we've done.
I hope this made sense, and if it's counted wrong on your test, please forgive me. I'm still fairly new to the Law of Sines.