Section 7.3: The Addition & Subtraction Formulas

In this section, we derived the six addition and subtraction formulas for sine, cosine, and tangent.

We started by finding the addition identity for sine, beginning with a triangle.

            ∆DFB:                          ∆ABC:                            ∆ABD:

             cos x = DF/DB              sin x = BC/AB              sin y = DB/AD

             DFL DB cos x               BC = AB sin x                DB = ADsin y

                                                                                          cos y = AB/AD

                                                                                            AB = AD cos y

So, we used these findings to figure out the sine of x + y (angle A). To find this, we started with:

Through substitution, we performed the following steps:

We then began to substitute more into the derivation:

By canceling out the AD, we end with:

sin(x+y) = sinycosx + cosxsiny

Here are all six of the addition and subtraction identities:

Sorry for my messy handwriting :)

These are all of the identities, found in similar ways as the sin addition identity. These are true for all values you plug in, because they are identities. 

I think that's about it for 7.3, see you all tomorrow!

- Jessica