First, a note on content, as this blog is still young and "developing" a style. It seems like I'm going to not only describe the process of learning trig, but I'm also going to be explaining what I'm learning in the simplest way I can. I'm doing this not because I want this to be an instructional blog - I'm very much a neophyte at this myself - but for several other reasons:
First, because reiterating what I'm learning in a longhand form, much like the practice scratchpad above, helps to cement these things in my mind. Second, because I know some of my readers are going to be math people, they can perhaps tell me when I'm going about something the wrong way. And finally, because I know some of my readers are decidedly not math people, but are curious about it - perhaps they'll pick up a tidbit here and there and begin some studies themselves.
With that in mind, today we're going to talk about radians.
2π+5 radians = Burning Down the House
(With apologies for the terrible joke in the section title.) As I mentioned before, in trigonometry (and in all the "advanced" mathematics, it seems) angles are expressed not in degrees, but in radians. So - what is a radian, and how does it relate to degrees? Learning the difference is not quite as neat as learning the difference between meters and feet - but it's similar. So let's start with what we know from geometry.
Anyway, the unit circle here is divided up into some common angles, expressed in degrees, much like we learned in geometry class. Half a circle is 180°. 1/4 a circle is 90°. 1/8 of a circle is 45°. And so on down the line - incidentally, these fractions will become more important in a future post where we'll talk about some newer developments in mathematics regarding tau - but for now, let's move on.
So this is a circle the way we learned it in geometry class. There are 360 degrees in one full turn. The conversion from degrees to radians is, at least at this point, fairly straightforward:
360 degrees = 2π radians.
At the left, the angle α = one radian. You can see the circle's radius expressed with r, and you can also see the arc subtended by the angle - the outside edge of the circle that lies within the angle's bounds - is also r. That means that the radius and that arc are... you guessed it. The same length. And because that arc and the radius are directly proportional, this is true no matter how large the circle is. That angle, in a circle of any size, will always be one radian.
And I can feel my own eye beginning to twitch the more I try to explain that - so let's move on.
The one you'll want to remember though is half that.
180 degrees = π radians
That is the ratio that you'll want to remember for trig functions. Because from that, you can extrapolate just how many radians are in one degree, and vice-versa, thus:
1 degree = π/180 radians
180/π degrees = 1 radian
In case that lost you - you simply take the number of the unit you want (degrees or radians) and divide both sides by it. One side will work out to 1 degree or 1 radian, and the other side will have your answer. If that completely boggles your mind, don't feel bad. It took me a little while to see it too.
So - back to our unit circle, since we see that 180 degrees = π radians, pretty much everything is expressed in how far it is from 180 degrees. 90 degrees is half that - so π/2 radians. 45 degrees is a quarter of 180, so π/4 radians. It gets a little more complicated once you get into the angles that are greater than 180, but that's essentially the idea.
So let's apply this to a little exercise. Here is a triangle:
Remember our mnemonic device: SOH CAH TOA. We need to find the adjacent side's length, and we're given the hypotenuse. So we're going to be using out cosine formula (from CAH): cosθ=adj/hyp, thus:
cosπ/4 = x/10√2
First of all, we isolate the variable x by multiplying both sides by 10√2:
cosπ/4(10√2) = (x/10√2)10√2
which leaves us with...
cosπ/4(10√2) = x
Now. The problem is we don't know what the cosine of π/4 is - we'll have to look that up (or maybe we just know it, since it's a 45 degree angle). There's no way to work it out from what we're given here, at any rate. The cosine of π/4 is, in fact, √2/2.So we have:
√2/2 (10√2) = x
Here's where it gets a little difficult to express here, since I can't write fractions straight out the way I would on paper. So I did it in GIMP. Remember, when multiplying fractions we simply multiply the numerators, then the denominators, then reduce. Since 10√2 isn't a fraction, it gets a 1 in the denominator. Here's how it works out:
Again - don't feel like a buffoon if this leaves your head spinning a little. That's why I have pages of scratch paper full of these functions just for practice. I just kept doing them until the light bulb came on.
That's all for now... wow, what a lengthy post. I think I'll allow you to digest and dissect - until next time, when we may take a step back from the nuts and bolts to talk philosophically about trigonometry for a bit. Thanks for reading as always!