WEBVTT
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Hi. Today we're going to talk about the unit circle
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and how we can relate it to trigonometric functions.
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The unit circle is a circle of radius 1, and it has the equation
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x squared plus y squared equals 1.
At least that's the simplest equation for it.
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Now we know that the circumference of the circle is equal to 2 pi r.
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But in the unit circle the radius is 1, so the circumference of a unit circle is 2 pi.
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This is going to help us when we define the trigonometric functions.
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Before we start defining the trigonometric functions using the unit circle,
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we're going to define the central angle to be t radians.
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Now you remember that you've learned already that the arc length,
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the length of any intercepted arc here, is equal to r times theta.
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But in the unit circle we have that r is equal to 1, and we're defining theta,
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the angle, to be t. So we have that s is equal to t.
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So in a unit circle, now this is not any circle, but in a unit circle
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the radian measure of the central angle is equal to the length of the intercepted arc.
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One thing I want to point out is that t can be any real number.
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You can go around this circle in either direction, positive or negative direction,
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and you can go around it as many times as you want.
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So we can define the trig functions using this because t is any real number.
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Now what you've already done in trig is you have defined, for general angles,
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we have defined the sine of theta to be the value y over r
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and the cosine of theta to be x over r.
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But in the unit circle, r is equal to 1. That's one of the nice things about it.
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It makes it so useful. So r is equal to 1, and now we can define
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the sine of t to just be the value y and the cosine of t to be the value x.
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Now remember this is just on the unit circle. This is not for anything.
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OK, so now we're able to define all six of the trigonometric functions
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using the unit circle. We have for any real number t,
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now remember it can be positive, it can be negative, it can be 0,
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and a point P (x,y) that corresponds to this real number t,
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we define our six trigonometric functions
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as the sine of t is equal to y, the cosine of t is equal to x,
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the tangent of t is equal to y over x but x cannot be zero.
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And then we have the reciprocal functions, cosecant of t is 1 over y, y not zero.
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The secant of t is 1 over x, this is the reciprocal, x not equal to 0,
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and the cotangent is the reciprocal of tangent, x over y with y not equal to zero.
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So you need to learn these six trigonometric functions
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as we proceed through the next examples.
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Now let's do an example. We're going to find the values of the
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trigonometric functions for t with the point negative 3/5, 4/5,
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which is on the unit circle. That's what's nice about the unit circle.
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It makes defining these functions very simple. We can write that
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the sine of t, now remember that we define the sine of t to be the value of y,
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so the sine of t is equal to 4/5.
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We define the cosine of t to be x, so the cosine of t is negative 3/5.
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The tangent is defined as y over x, and you might remember
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that's sine over cosine, same thing, and that would reduce to negative 4/3.
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And then you can find the reciprocal functions using the definitions.
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Anytime you're working with functions you need to understand and be aware
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of what the domain and range of the function is.
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We define the sine of t to be y, and we already talked about the fact that
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t can be any real number. You can go around the unit circle.
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You can go around in the positive direction, negative direction,
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so t can be any real number.
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So the domain which is the input for the function, that's what you can put
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into the function, is any real number. So the domain here is going to be all reals
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which we will just designate with the interval notation
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negative infinity to infinity. That's all reals.
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Now the range of the function, that's what comes out of the function.
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That's the number that the function can actually equal.
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And if you look at the unit circle, and in general this is true
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for all sine and cosine functions,
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the value that the sine function is going to take on, the value that you put
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a number in, you take the sine, you get a number out,
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that number is always going to be between one and negative one.
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So the range of the sine function is the interval negative one to one,
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and the same is going to be true for the cosine function.
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So the domain for the sine and the cosine function is all reals.
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The range for the sine and the cosine function is always
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between negative one and one inclusive.