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When you condense the idea of a circle down to its very essence, its most basic form—the ur-circle, or urcle if you will—you get the unit circle. It's a unit because it has a radius of 1.
That circle is crammed with more points than a winning game of Tetris. We'll need to know all these points to solve trig equations.
Don't go trying to memorize the whole unit circle just yet, though. We've got all kinds of tricks and tips that make working with the unit circle a cinch.
So, for now, you have our permission to let your eyes glaze over while looking at it. Not that you waited for permission, did you?
In preparation for this unit, the angles we know and love have taught themselves a new trick: to play fetch.
That may not sound impressive to you, but they're rather proud of themselves, so let's play along.
The starting line for the game is the right side of the x-axis of the graph. That is now ground zero, or 0 degrees. When we throw the stick, it lands on the edge of the unit circle. The angle rushes out counterclockwise around the circle.
The angles are still the same size as they've always been, though. For instance, the angles 90, 180, and 270 each fall on one of the axes when going clockwise around the circle. This is one of the few times that it's okay to fall on an axe.
Sometimes, just to shake things up, we'll have negative angles to deal with, like -200 degrees. They stomp around the unit circle in a bad mood, moving clockwise instead of counterclockwise.
Any positive angle can be given as a negative angle instead, and a negative angle has a related positive angle in the same spot as well. It's like having an evil twin, but with a lot less maniacal laughter.
The idea of using degrees to measure angles is somewhat arbitrary: the ancient Babylonians decreed that there were 360 degrees in a circle because it was convenient for them. Well, this isn't ancient Babylon, so that isn't good enough. We want to use a measure that's convenient for us.
It turns out that there is a more convenient measure to use for angles when working with trigonometry and calculus. The radian is a measure for angles based on the characteristics of circles. In particular, there are 2π radians in a full circle. Uh, 2π, 2π, where have we seen that before?
Why, the unit circle. (Bet you didn't see that one coming. No way, no how.)
The unit circle has radius 1, so its circumference is 2π.
Convert the following from degrees to radians: 90°, 180°, 270°.
We want to be able to switch between radians and degrees like it ain't no thang. The secret to doing so is realizing that 360 degrees and 2π radians are equal. That means that:
But shhhh! Don't tell anyone. It's a secret.
From here, the problem is a straightforward case of unit conversion. We like straightforward.
An angle of radians is the same as an angle of 90 degrees. This makes sense, because they are both of the angle of a full circle. What does not make sense is an 8-foot tall Wookie living on the planet Endor. It does not make sense.
An angle of π radians goes halfway around the circle. There are not a lot of people we would go to the other side of a circle for, but pi is one of them. (Besides, pi owes us 20 bucks.)
The three angles we've looked at so far—, π, and —all fall on the axes of the graph. So do 0 and 2π. (They're even on the same one together: how scandalous.) These angles will act like reference points for us in later sections, to help us keep our bearing when working with unfamiliar angles.
They'll be like the constellations were for sailors in ancient times, but with less scurvy or getting lost because we actually aren't that good at sailing.
Convert the following from radians into degrees: , and .
Maybe we took things a little too fast. Let's see if we can go back to degrees from radians. Just in case this whole "working with radians" thing doesn't work out.
One down. The next one is:
We're knocking these out in double time.
Woah, wait up: angles can be greater than 360 degrees? What would that even look like?
Oh. It makes a full circle and then keeps on spinning. 420 degrees is actually the same as a (420 – 360) = 60 degree angle on the unit circle; it's also the same as radians. When two angles have different measures but end up on the same spot on the unit circle, they're said to be coterminal.
That makes it sound like both angles are dying together, but it really means that they terminate (that is, end) at the same place.
Anyway, it turns out that working with radians isn't that bad. We just have to relearn the sizes of some common angles; is a right angle, 2π is a full circle, and so on. Once we do that, using radians is no harder than using degrees for angles.
All our fears were as unfounded as a twice-lost sock.
Get it? We found the sock once, but then we lost it again. We "unfound it." See, if you say it fast, it sounds like "unfounded." Th-that's funny, right? Okay, moving on.
What is love? Well, we've heard that, when the moon hits your eye like a big pizza pie, that's amore, which is like love but foreign. If the moon were made entirely out of pizza, it probably wouldn't have taken people until 1969 to get there. There probably wouldn't be much moon left, either.
When it comes to pizza, we're a big fan of a good crust. We're such big fans that we started measuring the length of all our crusts (don't judge), and we figured something out.
If we cut a pizza in half, then each half has half of the total crust. If we cut it into fourths, then each slice has ¼ the crust. No duh, right? Well, this rabbit hole goes deeper. And it's full of pizza.
A slice of pizza is cut with a 30° angle; the pizza has a 12 cm diameter. How many cm of crust is there?
Our first question is, what toppings are on this pizza? We're not eating any weird toppings, like caviar or pickles. Next we want to know how much crust is on the whole pizza. In math-speak, we're asking about a circle's circumference. That's C = 2πr, or 12π cm for our pizza.
Our last pizza puzzle piece is, how much of a pizza does 30° cover? Well, we can use change our angle into those new-fangled radians to figure that out. There are 360° in a circle, so it's:
That's not a lot. It's a mighty convenient number for us, though, because it means the length of our crust, what mathematicians would dryly refer to as the arc length, is π, or about 3.14, cm. Now that's what we call a pi crust.
In general, we can find the arc length by the formula . We multiply the circumference of the circle by the fraction of the circle taken by the angle x. Things are even easier when we're working with radians. Then our formula is:
L = x r
We just multiply the angle by the radius (not the diameter) to get the arc length. If only we could find our socks this easily.
One last tidbit before we move on. Notice that, if the radius is 1, then the arc length equals the angle in radians. That means an angle in radians is the same as the arc length of that angle on the unit circle. Some people use this as an alternate definition for radians, and those people are math nerds.
The crust isn't the only good part of a pizza. We like the rest of it too, so now we're going to find how much of it we've got. That's like a thinly veiled excuse to do some math, and it sounds like finding the area of a circle.
Just like we took the circumference and chopped it up to find the arc length, we're doing the same with the total area to find the area of a sector. That's just some portion of a full circle.
When we're in degrees, we just multiply the total area by the fraction of the circle taken by our angle. When in Rome, do as the Romans do. And when we're in radians, well, let's replace that 360° with 2π and find out what happens:
We get rid of the π, we divide by 2, and we're done. Now that's a formula we can use.
What is the area of a sector of a circle measuring 6 in across that has an angle of π?
We can just plug and chug into our formula.
Man, we are really hungry now for some reason. Maybe we'll order some Chinese food.
The three wise men of the unit circle are . They bring with them gifts of knowledge, good grades, and burritos. That's a sweet haul.
We want to know each angles' x and y coordinates on the unit circle. Why? We'll never teeeellll.*
Let's start off with the angle . It's a 45 degree angle in radian disguise. That mustache isn't fooling anyone, though.
We can find x and y using the special right triangle for a 45 degree angle. The ratio of the sides is . In our case the hypotenuse is 1, so we need to solve . (A right triangle with side length s will be a 45-45-90 right triangle if the sides are length .)
Both sides are equal, so we have the point for the angle . We still won't tell why we want this information*, but we have it.
Now let's figure out the other two angles. is 30 degrees, while is 60 degrees. Yeah, the 3's and 6's not matching up can be confusing. We tried telling them that, but those wise-aleck angles told us, "Division does not work that way."
We can take care of both angles with one triangle. A 30-60-90 right triangle (or perhaps we should say a right triangle) has a ratio of , corresponding to the sides opposite 30:60:90. The hypotenuse is 1, so:
That gives us for , and for . Now we can talk about what all this stuff means.
You didn't think we'd really hold out on you? Pshaw.
Take a look at the special right triangles. Notice anything interesting about them?
No, nothing? Can't say we blame you. How about you compare the sine and cosine of each angle to their points on the unit circle?
x = cos A
y = sin A
Can you feel your mind being blown? Cosine equals the x-coordinate for an angle, and sine equals its y-coordinate. By learning the points on the unit circle, we also learn sine and cosine for each angle.
That means we already know sine and cosine for and 2π. We're unlocking secret knowledge from our brains that we didn't know they had. Who knows what mutant power we'll develop next?
We'll be using all of these angles a lot from now on, so you definitely want to memorize their cosine and sine…or use this handy trick for keeping them straight. It's as easy as 1-2-3 1-2-3.
*We'll totally tell. Soon.
How will we find sine and cosine of these angles? Asking politely doesn't work, looking up the answers online is a no-go (…Shmoop doesn't count), and math problems are notoriously resistant to torture.
We already know some values for angles in the first quadrant, and we'll use that information to figure out other angles. The missing piece to this puzzle is the corresponding acute angle (sometimes called the reference angle).
They're the angles in green. They cling to the x-axis like a joey in its mother's pouch (adorable).
The important thing to note is that the sine and cosine of any angle are equal to the corresponding acute angle's—except for their signs. x and y change sign according to which quadrant they're in.
What are the sine and cosine of ?
No, its sine is not Aquarius, and it cannot cosine a loan for you either. Those homophones are a stretch, even for us.
Our first order of business is to find the corresponding acute angle. is a little less than π, so we'll use that as our guide.
Looking at the graph makes it obvious that the closest part of the x-axis has angle π. If we subtract our angle from the x-axis, we can find the difference between them:
Our corresponding angle is . From the last section, we know that both x and y are . So hey, problem over, right?
That means x is (which means so is cosine), and y (and also sine) is .
Now we're done. With this problem, at least. We have another lined up. Sorry.
What are the sine and cosine of ?
Yeah, we just switched around the 3 and the 4 from the last problem. It's not lazy, it's efficient.
Anyway, let's make another graph. Our angle is a little larger than π, but less than .
Now, some people would look at this and say, "The angle is closest to , so the corresponding acute angle must be ."
Those people would be wrong (as Dr. Cox never tires of telling us). The corresponding acute angle goes with the x-axis, not the y-axis; remember the cute kangaroo joey? It was adorable, and with the x-axis. We find the correct angle by subtracting π out from :
We're in the third quadrant, so both x and y are negative. That means and .
You've got to keep an eye on those negative signs. They're easy to lose, but they always cause trouble if they're forgotten somewhere.
One last thing to note. Angles can get big; really big. How would we find the corresponding acute angle for a ridiculous angle like 57π?
We'll answer that question with another one: 2π, or not 2π? (That is the question.) And the answer is always "2π." That's a full circle, so subtracting 2π from an angle doesn't change its position on the unit circle.
57π – 2π = 55π
55π – 2π = 53π
Just keep on going, until we hit:
3π – 2π = π
So 57π is in the same position on the unit circle as plain ol' π. We know sine and cosine of π, and they'll be the same as sine and cosine of 57π.
That's how we work with big angles: whittle them down to size until they are manageable. Whittle whittle whittle.
We just like saying "whittle" out loud.
Negative angles, like rebellious teenagers, are contrarian. "I don't have to listen to you, Dad. I don't have to go counterclockwise if I don't want to. Gawd!"
Not that we have anything against rebellious teenagers; we have a few stories from our younger days that would make your hair stand on end, ya whippersnapper. Being so contrary does tend to make negative angles predictable, though.
Cosine plays it cool. Positive, negative: cosine doesn't care, he'll just keep on doing what he wants. Sine, though, can't stand being told what to do. If you try to give orders to sine while there's a negative angle in the mix, sine will do the exact opposite of what you said.
See? The original angle (A) is in red, and the negative angle (-A) is in blue. The relationships between A and -A are the Negative Angle Identities (catchy name, right?):
cos (-A) = cos A
sin (-A) = -(sin A)
Mathematicians call cosine an even function, and sine an odd function, based on these identities. We'd call them Joe and Larry, but that isn't very descriptive.
Even functions are such that f(-x) = f(x), which means that putting in a negative value returns the positive value instead. Odd functions, though, return the opposite of the positive value; f(-x) = -f(x).
If we want to know the sine or cosine of a negative angle, we can express it in terms of the positive angle. This works for any angle of any size in any quadrant, any time and any where. If we say "any" any more times, we might run out of "any"s. And that wouldn't be any fun.
What are the sine and cosine of ?
Why do you always have to be so negative, ? Let's inverse that frown upside down.
Let's start Operation No More Sad Face by figuring out sine and cosine of positive
Phase 1 is now complete; scowl inversion is at 20%. is located in the fourth quadrant, so x is super smiling while y is down in the dumps. That means that:
We've achieved a 70% bad-mood reversal. Don't mistake this for true happiness, though. We've found sine and cosine for positive , but we wanted to find them for .
Now is the time to deploy our secret weapon, the Negative Angle Identities.
Remember that cosine is chill; he was while the angle was positive, so he stays that way while it's negative. Sine is another story.
We've done it. We have achieved maximum happy. S.o. h.a.p.p.y.
The easiest mistake to make with negative angles is to stop too early and not actually apply the Negative Angle Identities. You can tell if you forgot by the aching sadness in your heart.
You know what would make us really happy? Jetpacks. We'd fly around fighting crime, and bears, and criminal bears. Woosh. Like a superhero, but Shmoopier.
Anyway, that's pretty tangential to what we're here to talk about, which is the tangent of different angles. We've also got cosecant, secant, and cotangent to do as well. Luckily, they all work the same way. If they worked different ways, that'd be four times the work, and we'd honestly flake out before finishing.
The key to finding angles for these functions is to know that they're all knock-off brand soda to sine and cosine's Coke and Pepsi.
How about we solve a problem? We're thinking….
The angle is a little less than π, so it's in the second quadrant.
That's our corresponding acute angle. We need to find its cotangent. Looking back at that list we made just a minute ago, we see that:
We stare at our left hand for a little bit, and discover that cos is and sin is .
That means the value of cot is , but now we need its sign. We're dividing cosine by sine, so in the second quadrant we're dividing a negative (cosine) by a positive (sine). If trig functions were magnets, we'd have a hard time pulling them apart now. They aren't, though, so instead we have:
In order for cotangent (or tangent) to be positive, both cosine and sine must be either positive or negative together, like in Quadrants I and III. Quadrants II and IV are negative. Quadrant V exists outside of our dimension, and constantly produces ice cream and volcanoes. They tend to balance each other out.
Cosecant and secant have the same sign as sine and cosine, respectively. We don't need a whole new sign language to work with them.
There is a danger to finding the angles of trig functions. All four trig functions other than sine or cosine involve division, so there is the risk of the ever-dreaded division by zero. Tangent, for instance, would divide by zero whenever cosine equals zero, at 0 and π radians. At those spots, the function is undefined, like the sound of one hand clapping.