We know the exact value of all the trig functions for 
, because of the special triangles. We also know the values at each axis. But we need to know more, and we need to do it without a calculator.
We're going to start with the addition formula for sine. It's not as easy as just adding some numbers together; this is trigonometry, and that makes it more complicated than usual. So, take our hand as we walk through the proof for the formula together. It might get a little scary, but knowing where the formula comes from will help us understand why it looks the way it does.
Let's look at a picture of two angles we want to add together, hanging out in two separate triangles.
We can find sin(α + β) using these triangles…almost. Let's add one more line in there to help us out:
Okay, so we added two lines, close enough. We made the triangle ADE, with ∠DAE measuring α + β, and then a few more triangles towards the top that will be useful.
Now that the figure is in place, let's start poking it to see what falls out. Luckily, there is a 0% chance of any bees, wasps, or hornets coming out of a math problem like this. See, proofs aren't all bad; they could be a lot worse, and much more painful.
Let's look at ∠CAB (which we know is α) and ∠FCA. These look an awful lot like alternate interior angles, because, hey, they are. Being alternate interior angles, they have the same measurement, so we know that ∠FCA is also α.
Where are we going with this? We're going to Disney World, someday. We're also going to triangle FDC. ∠FCD must be 90° – α, because together ∠FCD and ∠FCA are a 90° angle. That's important because of ∠FDC. We know all angles in a triangle add up to 180°, so:
∠FDC + ∠FCD + 90° = 180°
∠FDC + (90° – α) + 90° = 180°
A little bit of algebra and bing-o bang-o, ∠FCD is also α. Let's put those in our diagram. Mostly because we're tired of scrolling up so much every time we forget which angle is which.
To find sin(α + β) we can use a trig ratio from triangle ADE:
Seems easy enough, right? Problem is, we don't know the lengths of all of these new lines. In fact, we don’t know nothing about any lines. What's a know-nothing to do in a situation like this? We don't know, but we're going to do something with all of these angles we have sitting around.
That means using more trig functions, and that means we need to break up some of our lines into smaller lines:
Here's something tricky we can do: replace 
with it's equal on the other end of the rectangle, 
.
Now that we have broken this up, let's revisit our formula:
The lines 
and 
both live in triangles with an α angle in them. We'll ask their roommates to help us out with a little bit of trig.
Thanks for inviting us into your homes, but we need to rearrange things a bit so that we can continue this problem. Let's put the couch along the other wall, put the TV next to the bookcase, and isolate 
and 
so we can replace them in our formula.
With that out of the way, let's stick these into 
We're almost there. Stick with us. The lines 
are all parts of the triangle DAC. And that triangle has the angle β in it. Cha-ching, we've hit paydirt. We can replace the ratios in the formula with trig functions using β and rearrange it a bit to get:
sin (α + β) = sin (α) cos (β) + cos (α) sin (β)
We've done it; we've proven this formula to be true. Now we just need to be able to remember it when we work on our problems. Let's see, sine and cosine each show up twice, once with alpha and once with beta. They never share the same angle, angle. In the formula, sine comes first. Okay, that should help us remember.
Sample Problem
Find the exact value of sin (75°).
Put that calculator away, bucko. We didn't go through that whole proof to give up at the finish line We can split 75 into (30 + 45), and from there the solution is more obvious than George Washington's nose on Mount Rushmore. And definitely more obvious than if we had to split this up as 
.
Let's use that addition formula:
sin (45° + 30°) = sin (45°)cos (30°) + cos (45°)sin (30°)
This we can do.
Pluga-chuga, pluga-chuga.
Okay, we need a break. We'll do cosine and tangent next time. We're going to put this on and relax for a while.