Which pair of angles in the correct category to indicate whether the pair of angles is pair of vertical angles, a pair of line angles or neither .

We can look at the full aspect of the triangle with one or two lengths missing

We call the full aspect T for triangle.

We know that half of c when we add sin to c = 1/2 of b when we add sin to b

We also know if we are adding sin to c we are also adding the opposite to it to so it looks like this as you draw a line of the angle you want to find.

T= 1/2b  (c sin A) = 1/2 c (a sin b) = 1/2 a(b sin c)

The picture shows the opposites and As you get used to typing in Sin it can show you another length if you use 2 lengths, and it can show you the angle if you use two lengths, and it can show you either angles if you change the side you use it will always be opposite. As Sin = Soh = opposite/hypotenuse.

This means you are looking for the adjacent side, but from the hypotenuse perspective we are looking for the opposite or the adjacent.

depending what way the triangle is up.

I call it opposite as they usually ask for the Side or base and give the lower angle so we are working with hypotenuse as an entry and any other side or lower angle to find the side or the base missing measure.

When we look for this side we usually enter an angle or both sides and it will confirm it, you only need to write down an example out of any maths book and test this.

Here is a picture to show you why opposites are used for ABC points.

We label the triangle and look for what letter the side will be opposite and enter that into the calculator. We ignore the angle side that is not being asked for.

As we only need the other angle if we are asked or if we need to try using cosine for the same triangle.

There is more to learn about Sin for circles its just another way to find an angle or a length.

As shown in the second attachment;

Let there be a circle where where R is the radius of the circle, angles C and D have the same angle. We ignore the R for a bit and prove that D = C and here as there are a smaller triangles inside a circle that we added being D we can prove by showing and using Sin.

Sin D = sin C = c/2R

This means we found equal lengths.

a/Sin A  = b/Sin B = c/Sin C = 2R

So we can use Sin to express how the calculator changes A -B- C for a triangle without it we are only left with mind map of ratios that are not precise calculations for triangle lengths as they do not represent the angles.