Starting with the Cosine Difference Formula

We are on our way to getting a whole fabulous collection of trigonometric identities, and where it all starts is with this problem.

Problem: If I know trig functions for two different values or angles, is there a way to use this to figure out the trig functions for the sum or difference of these values.

Now at first sight it is tempting to say something like sin(A+B)=sinA+sinB, I suppose because this looks like something you do in algebra all the time. Isn't that what you do in algebra when you see a parentheses and something on the outside of it? No. It sounds good, but it is total nonsense. This is not a general rule for dealing with things outside of parentheses. What you are trying to use when you do that is the distributive property and the distributive property is only a rule about multiplying and adding not a general rule for dealing with parentheses and the sin is not being multiplied by the A+B. It is a function that is being applied to it. If you look at a picture on the unit circle of what is going on here you can see that this is total nonsense.

The blue line here represents sinA, the red line represents sinB, and the green line represents sin(A+B). Clearly the length of the green line is not the sum of the lengths of the blue line and the red line. In fact from looking at this picture it is not at all clear that there should be any way of relating sin(A+B) and sinA and sinB. There is, but we have to resort to quite a clever trick in order to figure it out. It turns out that the easiest identity of this sort to figure out is the one for cos(A-B) and then after we get that one others can be derived from it.

Cosine Difference Formula

To figure out the identity for cos(A-B) you need to draw the right picture and from this picture and the rest is just patience. I used to tell students that it just fall out once you have the right picture, but most don't buy the  falling out part and have claimed that it is more like giving birth. But still once you have this one trick the inspiration part is done and all that is left is perspiration. Equal angles have equal length chords, so the red line is the same length as the blue line. Believe it or not, if we play with this fact enough, it will produce the identity we are looking for. According to the distance formula, the square of the length of the blue line is and the square of the length of the red line is Setting these equal we get Then if we subtract 2 from both sides and divide both sides by -2 we get and there is it just coming out of the middle of nowhere, just the kind of identity we were looking for, just like magic it appeared in front of our eyes.

More Identities

Now this cosine difference identity is kind of a treasure chest, because you can get whole bunches of other identities out of it.

The first thing to do is get a cos sum identity from it. To add, change the signs and subtract. Isn't that what they taught you when you learned about plus and minus numbers? Well, not quite, it was probably the other way around, but this way it works too.

Now we can write the sum and difference formulas for cosine as one equation like this. Next we can derive identities for sine from it. For this we need the cofunction identities. The idea is to write sine in terms of cosine. We could do it by using the first of the Pythagorean Identities, but that is too complicated. The way that works best for this is to use some identities that you may have noticed when we first introduced sine and cosine in terms of right triangles, which has to do with complementary angles. Complementary angles sit on the same right triangle, because the sum of the angles of a triangle is 180, so the cofunctions switch for them, because the one angle's opposite is the other angle's adjacent and vice versa. In terms of radian measure this says You can also see this on the unit circle by looking at this picture. Notice that the x coordinate for theta is the same as the y coordinate for pi/2-theta and vice versa. Pi/2-theta is the reflection across the y=x line of theta.

Replacing theta with A+B will help us find a sum formula for the sine function. From this we get

And now all we have to do is apply this to A and -B to get a difference formula for sine. Then just like with the cosine, we can write these together in one formula like this.

Practical Uses

One of the things you can use this for is to find exact values for sines and cosine of sums and differences of your special angles like 75=45+30 or 15=45-30. There are exercises in many trigonometry books where you have to do that. What you need to do for this is write your angles in terms of the familiar ones like this and then apply the appropriate sum or difference formula and then plug in the familiar values.

I should probably include some examples of this, so if you have one that you are having trouble with, please let me know, and I will try to help you with it and maybe write it up here.

Tangent

Now for the tangent function. Didn't I tell you this was a treasure chest identity? Tangent can be written in terms of sine and cosine so once we have identities for sine and cosine, we should be able to get one for tangent from them. So here we go. This is fine, but with a little trick we can make it nicer, by getting it all in terms of tangents. The trick is to divide top and bottom by cosAcosB. If you think about it this makes sense, because at least you know it will put cosines in the denominators like tangent has.

Summary

Putting this all together we have these three sum and difference formulas for sine, cosine, and tangent. So I will display for you these formulas for you now. In maybe trigonometry class, including mine, you get formula sheets for these, so if you forget them you will be able to look them up. But still you should memorize them, because you won't be able to quickly recognize them for proving identities or solving equations if you don't. Also if you go on to calculus you might not get a formula sheet for them there. I usually find a good way to memorize things like this is play around with them a bit. You will gradually get them memorized just by using them, but it pays to be a bit more active about it than that. If you look at them together you can see some patterns.

The way I always remember the sine and cosine identities is this. First I remember that they both have all possible combinations, sinA,sinB,cosA,cosB, and these only occur once. If you write them always in one order they go AB AB. And then there is this curious thing about them. They are both messed up in some way and neat in another way. For cosine, the cosines are together and the sines are together, but for sine they are scrambled. As soon as I know that they are scrabbled, though I know exactly what order they have to be in because first I remember that because it is sine, it starts with sine and then after that they are in the only arrangement that they can be in so that all of them are included and included exactly once. But way of compensation, as if the mathematical gods were trying to be fair, for sine the plus and minuses are neatly lined up, and for cosine they are reversed. Then also the other thing you can do to remember them is keeping in mind that it is AB AB, you remember the rest of it in a kind of abbreviated style by thinking of cosine as coscos sinsin and sine as sincos cossin. (This sounds better spoken that it looks written.) At the end of the day, though, you will probably remember them best by the way you invent, so just play with them a bit. For the tangent I don't really have any advice, but it does help to remember that tangent is sine over cosine and if worse comes to worse it isn't that hard to derive it for yourself if you just remember the trick about dividing top and bottom by coscos.

Reduction Formulas

Another thing you can get from the sum and difference formulas is a number of formulas for what happens when you push theta around the circle a bit. You can get any of this also by looking at the unit circle as well, but the sum and difference formulas give you an extra check on them and method that you can use to figure them out if you are feeling unable to think on an exam. The general rule for these sorts of identities is that for adding or subtracting odd multiples of pi/2 the co's switch just like cofunction identities and for even multiples (multiples of pi) you get the same function, and in either case some have minus signs and some don't, and that part is always the tricky part to remember.  I never remember which one gets the minus sign and I have been teaching this class for over 10 years, so I don't see any point in you memorizing it. There are several ways of getting it quickly when you need it.
 

The Unit Circle

This is my preferred method. Just position your theta someplace on the unit circle, preferably in the first quadrant, since it is simplest, and do the necessary rotation and see what happens. For example you can draw this picture for theta + and - pi/2. Remember, you know it will switch the co's, so all you are looking for is the sign.

Even and Odd

Another way is to derive them by combining the first cofunction identities with the even and odd properties of sine and cosine and the identities you learned earlier about rotations involving pi around the unit circle. This is probably the simplest rigorous way you can do it, but it is a but fussy.

Using the Sum and Difference Formulas

This way is kind of cracking a walnut with a sledge hammer, but it works very dependably and also gives you another application for these identities. In fact these sum and difference formulas are very powerful in this way. Once you know them, you are safe from forgetting all these various identities involving rotations around the unit circle, because you can get them all as special cases of the sum and difference formulas. Here are a couple of examples. Do watch out, though, if you do this for tangent, do it by writing tangent in terms of sine and cosine, not by the tangent sum or difference formula, because tangent is undefined at pi/2.

I haven't included very many examples in this, so please feel free to ask questions about one if you have them, and perhaps others can benefit as well as you.

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