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Get ready for Precalculus
Course: Get ready for Precalculus > Unit 5
Lesson 4: Unit circle introductionUnit circle
Learn how to use the unit circle to define sine, cosine, and tangent for all real numbers. Created by Sal Khan.
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Do these ratios hold good only for unit circle? What if we were to take a circles of different radii?(183 votes)
The ratio works for any circle. The advantage of the unit circle is that the ratio is trivial since the hypotenuse is always one, so it vanishes when you make ratios using the sine or cosine.(330 votes)
What is the terminal side of an angle?(114 votes)
straight line that has been rotated around a point on another line to form an angle measured in a clockwise or counterclockwise direction(25 votes)
So how does tangent relate to unit circles? And what is its graph?(17 votes)
I think the unit circle is a great way to show the tangent. While you are there you can also show the secant, cotangent and cosecant. I do not understand why Sal does not cover this.
Using the unit circle diagram, draw a line “tangent” to the unit circle where the hypotenuse contacts the unit circle. This line is at right angles to the hypotenuse at the unit circle and touches the unit circle only at that point (the tangent point). Extend this tangent line to the x-axis. The distance of this line segment from its tangent point on the unit circle to the x-axis is the tangent (TAN). If you extend the tangent line to the y-axis, the distance of the line segment from the tangent point to the y-axis is the cotangent (COT). To determine the sign (+ or -) of the tangent and cotangent, multiply the length of the tangent by the signs of the x and y axis intercepts of that “tangent” line you drew. For example, If the line intersects the negative side of the x-axis and the positive side of the y-axis, you would multiply the length of the tangent line by (-1) for the x-axis and (+1) for the y-axis. You will find that the TAN and COT are positive in the first and third quadrants and negative in the second and fourth quadrants.
As a bonus, the distance from the origin (point (0,0)) to where that tangent line intercepts the x-axis is the secant (SEC). The sign of that value equals the direction, positive or negative, along the x-axis you need to travel from the origin to that x-axis intercept. The distance from the origin to where that tangent line intercepts the y-axis is the cosecant (CSC). The sign of that value equals the direction positive or negative along the y-axis you need to travel from the origin to that y-axis intercept.
Some people can visualize what happens to the tangent as the angle increases in value. When the angle is close to zero the tangent line is near vertical and the distance from the tangent point to the x-axis is very short. As the angle nears 90 degrees the tangent line becomes nearly horizontal and the distance from the tangent point to the x-axis becomes remarkably long. You can, with a little practice, “see” what happens to the tangent, cotangent, secant and cosecant values as the angle changes.
The second bonus – the right triangle within the unit circle formed by the cosine leg, sine leg, and angle leg (value of 1) is similar to a second triangle formed by the angle leg (value of 1), the tangent leg, and the secant leg. In this second triangle the tangent leg is similar to the sin leg the angle leg is similar to the cosine leg and the secant leg (the hypotenuse of this triangle) is similar to the angle leg of the first triangle. When you compare the sine leg over the cosine leg of the first triangle with the similar sides of the other triangle, you will find that is equal to the tangent leg over the angle leg. Therefore, SIN/COS = TAN/1 . You can also see that 1/COS = SEC/1 and 1^2 + TAN^2 = SEC^2. A bunch of those almost impossible to remember identities become easier to remember when the TAN and SEC become legs of a triangle and not just some ratio of other functions. The angle line, COT line, and CSC line also forms a similar triangle.
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When you graph the tangent function place the angle value on the x-axis and the value of the tangent on the y-axis. At the angle of 0 degrees the value of the tangent is 0. At 45 degrees the value is 1 and as the angle nears 90 degrees the tangent gets astronomically large. You are left with something that looks a little like the right half of an upright parabola. At negative 45 degrees the tangent is -1 and as the angle nears negative 90 degrees the tangent becomes an astronomically large negative value. This portion looks a little like the left half of an upside down parabola. This pattern repeats itself every 180 degrees. Do yourself a favor and plot it out manually at least once using points at every 10 degrees for 360 degrees. It may not be fun, but it will help lock it in your mind. Give yourself plenty of room on the y-axis as the tangent value rises quickly as it nears 90 degrees and jumps to large negative numbers just on the other side of 90 degrees.(133 votes)
- I hate to ask this, but why are we concerned about the height of b? What is a real life situation in which this is useful? Graphing sine waves?(17 votes)
- Say you are standing at the end of a building's shadow and you want to know the height of the building. you only know the length (40ft) of its shadow and the angle (say 35 degrees) from you to its roof. you could use the tangent trig function (tan35 degrees = b/40ft)
40ft * tan35 = b
28ft = b
Now you can use the Pythagorean theorem to find the hypotenuse if you need it.(30 votes)
- In the concept of trigononmetric functions, a point on the unit circle is defined as (cos0,sin0)[note - 0 is theta i.e angle from positive x-axis] as a substitute for (x,y). This is true only for first quadrant. how can anyone extend it to the other quadrants? i need a clear explanation... I think trigonometric functions has no reality( it is just an assumption trying to provide definition for periodic functions mathematically) in it unlike trigonometric ratios which defines relation of angle(between 0and 90) and the two sides of right triangle( it has reality as when one side is kept constant, the ratio of other two sides varies with the corresponding angle).... i think mathematics is concerned study of reality and not assumptions.... how can you say sin 135*, cos135*...(trigonometric ratio of obtuse angle) because trigonometric ratios are defined only between 0* and 90* beyond which there is no right triangle... i hope my doubt is understood..... if there is any real mathematician I need proper explanation for trigonometric function extending beyond acute angle.(14 votes)
[cos(θ)]^2+[sin(θ)]^2=1 where θ has the same definition of 0 above.
This is similar to the equation x^2+y^2=1, which is the graph of a circle with a radius of 1 centered around the origin. This is how the unit circle is graphed, which you seem to understand well.
Based on this definition, people have found the THEORETICAL value of trigonometric ratios for obtuse, straight, and reflex angles. This value of the trigonometric ratios for these angles no longer represent a ratio, but rather a value that fits a pattern for the actual ratios. I hope this helped!
Proof of [cos(θ)]^2+[sin(θ)]^2=1:
https://www.khanacademy.org/cs/cos2sin21/6138467016769536(6 votes)
This seems extremely complex to be the very first lesson for the Trigonometry unit. He keeps using terms that have never been defined prior to this, if you're progressing linearly through the math lessons, and doesn't take the time to even briefly define the terms. No question, just feedback.(10 votes)
The problem with Algebra II is that it assumes that you have already taken Geometry which is where all the introduction of trig functions already occurred. So Algebra II is assuming that you use prior knowledge from Geometry and expand on it into other areas which also prepares you for Pre-Calculus and/or Calculus. So if you need to brush up on trig functions, use the search box and look it up or go to the Geometry class and find trig functions.(19 votes)
- At2:34, shouldn't the point on the circle be (x,y) and not (a,b)? [Since horizontal goes across 'x' units and vertical goes up 'y' units--- A full explanation will be greatly appreciated](6 votes)
It would be x and y, but he uses the letters a and b in the example because a and b are the letters we use in the Pythagorean Theorema²+b² = c²and they're the letters we commonly use for the sides of triangles in general. It doesn't matter which letters you use so long as the equation of the circle is still in the forma²+b² = 1.(18 votes)
- You should at least explain what sin and cosine are. This is intro to trigonometry(7 votes)
sin is opposite/hypotnuse. cosine is adjacent/hypotnuse(5 votes)
- if someone is having hard time of understanding this, I would encourage you to get additional information from "organic chemistry tutor" YouTube channel(8 votes)
- What's the standard position?(5 votes)
A "standard position angle" is measured beginning at the positive x-axis (to the right). A positive angle is measured counter-clockwise from that and a negative angle is measured clockwise. (It may be helpful to think of it as a "rotation" rather than an "angle".)(6 votes)
2:34Video transcript
What I have attempted to
draw here is a unit circle. And the fact I'm
calling it a unit circle means it has a radius of 1. So this length from
the center-- and I centered it at the origin--
this length, from the center to any point on the
circle, is of length 1. So what would this coordinate
be right over there, right where it intersects
along the x-axis? Well, x would be
1, y would be 0. What would this
coordinate be up here? Well, we've gone 1
above the origin, but we haven't moved to
the left or the right. So our x value is 0. Our y value is 1. What about back here? Well, here our x value is -1. We've moved 1 to the left. And we haven't moved up or
down, so our y value is 0. And what about down here? Well, we've gone a unit
down, or 1 below the origin. But we haven't moved
in the xy direction. So our x is 0, and
our y is negative 1. Now, with that out of the way,
I'm going to draw an angle. And the way I'm going
to draw this angle-- I'm going to define a
convention for positive angles. I'm going to say a
positive angle-- well, the initial side
of the angle we're always going to do along
the positive x-axis. So you can kind of view
it as the starting side, the initial side of an angle. And then to draw a positive
angle, the terminal side, we're going to move in a
counterclockwise direction. So positive angle means
we're going counterclockwise. And this is just the
convention I'm going to use, and it's also the convention
that is typically used. And so you can imagine
a negative angle would move in a
clockwise direction. So let me draw a positive angle. So a positive angle might
look something like this. This is the initial side. And then from that, I go in
a counterclockwise direction until I measure out the angle. And then this is
the terminal side. So this is a
positive angle theta. And what I want to do is
think about this point of intersection
between the terminal side of this angle
and my unit circle. And let's just say it has
the coordinates a comma b. The x value where
it intersects is a. The y value where
it intersects is b. And the whole point
of what I'm doing here is I'm going to see how
this unit circle might be able to help us extend our
traditional definitions of trig functions. And so what I want
to do is I want to make this theta part
of a right triangle. So to make it part
of a right triangle, let me drop an altitude
right over here. And let me make it clear that
this is a 90-degree angle. So this theta is part
of this right triangle. So let's see what
we can figure out about the sides of
this right triangle. So the first question
I have to ask you is, what is the
length of the hypotenuse of this right triangle that
I have just constructed? Well, this hypotenuse is just
a radius of a unit circle. The unit circle
has a radius of 1. So the hypotenuse has length 1. Now, what is the length of
this blue side right over here? You could view this as the
opposite side to the angle. Well, this height is
the exact same thing as the y-coordinate of
this point of intersection. So this height right over here
is going to be equal to b. The y-coordinate
right over here is b. This height is equal to b. Now, exact same logic--
what is the length of this base going to be? The base just of
the right triangle? Well, this is going
to be the x-coordinate of this point of intersection. If you were to drop
this down, this is the point x is equal to a. Or this whole length between the
origin and that is of length a. Now that we have
set that up, what is the cosine-- let me
use the same green-- what is the cosine of my angle going
to be in terms of a's and b's and any other numbers
that might show up? Well, to think
about that, we just need our soh cah toa definition. That's the only one we have now. We are actually in the process
of extending it-- soh cah toa definition of trig functions. And the cah part is what
helps us with cosine. It tells us that the
cosine of an angle is equal to the length
of the adjacent side over the hypotenuse. So what's this going to be? The length of the
adjacent side-- for this angle, the
adjacent side has length a. So it's going to be
equal to a over-- what's the length of the hypotenuse? Well, that's just 1. So the cosine of theta
is just equal to a. Let me write this down again. So the cosine of theta
is just equal to a. It's equal to the x-coordinate
of where this terminal side of the angle
intersected the unit circle. Now let's think about
the sine of theta. And I'm going to do it in-- let
me see-- I'll do it in orange. So what's the sine
of theta going to be? Well, we just have to look at
the soh part of our soh cah toa definition. It tells us that sine is
opposite over hypotenuse. Well, the opposite
side here has length b. And the hypotenuse has length 1. So our sine of
theta is equal to b. So an interesting
thing-- this coordinate, this point where our
terminal side of our angle intersected the
unit circle, that point a, b-- we could
also view this as a is the same thing
as cosine of theta. And b is the same
thing as sine of theta. Well, that's interesting. We just used our soh
cah toa definition. Now, can we in some way use
this to extend soh cah toa? Because soh cah
toa has a problem. It works out fine if our angle
is greater than 0 degrees, if we're dealing with
degrees, and if it's less than 90 degrees. We can always make it
part of a right triangle. But soh cah toa
starts to break down as our angle is either 0 or
maybe even becomes negative, or as our angle is
90 degrees or more. You can't have a right triangle
with two 90-degree angles in it. It starts to break down. Let me make this clear. So sure, this is
a right triangle, so the angle is pretty large. I can make the angle even
larger and still have a right triangle. Even larger-- but I can never
get quite to 90 degrees. At 90 degrees, it's
not clear that I have a right triangle any more. It all seems to break down. And especially the
case, what happens when I go beyond 90 degrees. So let's see if we can
use what we said up here. Let's set up a new definition
of our trig functions which is really an
extension of soh cah toa and is consistent
with soh cah toa. Instead of defining cosine as
if I have a right triangle, and saying, OK, it's the
adjacent over the hypotenuse. Sine is the opposite
over the hypotenuse. Tangent is opposite
over adjacent. Why don't I just
say, for any angle, I can draw it in the unit circle
using this convention that I just set up? And let's just say that
the cosine of our angle is equal to the x-coordinate
where we intersect, where the terminal
side of our angle intersects the unit circle. And why don't we
define sine of theta to be equal to the
y-coordinate where the terminal side of the angle
intersects the unit circle? So essentially, for
any angle, this point is going to define cosine
of theta and sine of theta. And so what would be a
reasonable definition for tangent of theta? Well, tangent of theta--
even with soh cah toa-- could be defined
as sine of theta over cosine of theta,
which in this case is just going to be the
y-coordinate where we intersect the unit circle over
the x-coordinate. In the next few videos,
I'll show some examples where we use the unit
circle definition to start evaluating some trig ratios.