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You know those commercials that show those poor, sad, abandoned animals just looking for a home? Well, in Trigonometry Land they could have those same commercials for poor, sad, forgotten angles.
The special triangles and their angles have been getting all of the love so far, so it's time to branch out and include more. With the addition and subtraction formulas, we can reach out to a bunch of those previously abandoned angles (those poor things) that we never knew how to work with without our handy dandy calculators.
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.
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.
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.
More proofs? Couldn't we get a root canal, or listen to babies crying on loop for 10 hours instead?
We could do those, but they'd be a lot more painful than this will be. We did all the heavy lifting last time; finding cosine follows the same logic as sine, we just use different lines in our answer, which changes the final formula a little bit.
Once again, we start with:
As before, we'll play match-breaker to the numerator, breaking it up and shuffling the pieces around.
This time we have some subtraction, but it is the same idea. Now we can relate and to the angle α using trig functions.
Now we rearrange these functions for the lines, and stick them back into our formula equation. We've got déjà vu going on in here.
One more set of substitutions with β and we have the addition formula for cosine.
cos (α + β) = cos (α) cos (β) – sin (α) sin (β)
Think of sine and cosine as rival softball teams. Each angle sticks with their team—sine with sine, cosine with cosine—even if they have friends on the other team. And sine is having a very bad year; they've somehow scored negative points throughout the season. No one is even sure how they did it.
Find the exact value of cos (-135°).
Don't panic at that negative angle. We've got cosine, and cosine is an even function f(-x) = f(x).
Our answer will be the same as cos 135°. See, no need to panic.
There are a few ways we can get to 135°. Using our special triangles, we can use 135° = 30° + 45° + 60°, which would involve using the addition formula twice. That doesn't sound like so much fun, but there is another option.
135° is also equal to 90° + 45°. Taking a quick peek at the graphs of y = sin x and y = cos x, or at the unit circle, shows that sin (90°) is 1 and cos (90°) is 0. That's easy enough to work with, so let's use 90° + 45°
cos (90 + 45) = cos (90) cos (45) – sin (90) sin (45)
This formula might be even easier to use than sine's.
We still need to find the addition formula for tangent. It gets easier from here, though, we promise. Usually tangent makes things harder for us, but maybe it's warming up to us? The trick of it is that tangent is always equal to sine over cosine.
Like a butterfly bursting forth from its cocoon, we'll substitute in the addition formulas for sine and cosine to get a beautiful new equation.
Okay, it's not a beautiful butterfly yet. More like a sleepy wasp. We can fix this, though. We can cancel some terms, and trade in some of our sin α's, by dividing everything by cos α.
That's looking a little better, but it could use some more help. Let's divide the top and bottom by cos β now:
Maybe not all formulas can be butterflies. But at least we can use this one to find more angles.
Math Wars: The Radians Strike Back. And this time, it's brought a variable with it. We can still apply the formula, though.
Now we just take a special consultation call with the 30-60-90 triangle to simplify as much as we can. And unlike calling a psychic hotline, it doesn't cost us $4.99 a minute.
This may look more complicated, but it is actually simplified. We can't really do much with a trig function when it has addition stuck inside it.
We've got three formula-la-las for adding angles together.
If your brain is completely strapped for space, remembering the sine and cosine addition functions are the most important. We can find the tangent addition formula (and some others, shhh) using them.
But really, you must have some room left, or some spare clutter you can toss out. Do you really need to be able to quote the entirety of The Princess Bride forwards and backwards? (Actually, yes we do.)
Once we know the addition formulas, the subtraction formulas are a cinch. We just plug in a negative angle and watch the signs.
sin (α – β) = sin (α + (-β))
= sin (α) cos (-β) + cos (α) sin (-β)
Don't be such Negative Nancy's, sine and cosine. We'll use our knowledge of periodicity to pull those negative signs out of the trig functions.
Cosine is an even function, so a negative angle doesn't affect the sign. cos (-x) = cos (x)
Sine is an odd function, so a negative sign is pulled out to the whole function. sin (-x) = -sin (x)
sin (α – β) = sin (α) cos (β) – cos (α) sin (β)
This formula is just like the addition formula, except the sign in the middle is switched around. That is mighty convenient and easy to remember. Wouldn't it be nice if cosine and tangent worked the same way?
cos (α – β) = cos (α) cos (-β) – sin (α) sin (-β)
cos (α – β) = cos (α) cos (β) + sin (α) sin (β)
Oh! They do work the same way. The middle term switches sign each time. That is nice.
Find the exact value of cos (15°).
With 15° being so small, it should be pretty obvious that we need a subtraction formula for this one. We could do 15° = 60° – 45° or 15° = 45° – 30°. Either one would work, and neither one would be faster than the other. Let's work with *flips coin* 45° – 30°.
cos (45 – 30) = cos (45) cos (30) + sin (45) sin(30)
We wish we felt half as special as these angles do.
And now we can add and subtract angles with ease. Well, as long as we know the formulas.