Unit circle definition of trig functions Extending SOH CAH TOA so that we can define trig functions for a broader class of angles
Unit circle definition of trig functions
- 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 one
- so this length here from the center, and I've centered it at the origin,
- this length, from the center to any point on the circle is of length one.
- so what would this co-ordinate be right over there? Right where it intersects along the x axis?
- well X would be one, Y would be zero (1,0)
- what would this co-ordinate be up here?
- well we've gone one above the origin, but we haven't moved to the left or the right...
- so our X value is zero, our Y value is one.
- what about back here?
- well here our X value is negative-one, we've moved one to the left
- and we haven't moved up or down so our Y value is zero
- and what about down here?
- well, we've gone one unit down (one below the origin)
- but we haven't moved in the X direction... so our X is zero, and our Y is negative-one.
- now with that out of the way, I'm going to draw an angle...
- and the way I'm going to draws 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 draw along the positive X axis.
- so this is the... you can kind of view it as the starting side of the angle (the initial side of an angle)
- and then to draw a positive angle the terminal side we going to move in a counter-clockwise direction
- so a positive angle- means we're going counter clockwise
- ...counter clockwise...
- 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 that a negative would move in a clockwise
- ...clockwise... direction.
- so let me draw a positive angle
- a positive angle might look, something like this. (this is the initial side)
- and then from that I go in a counter-clockwise direction until I measure out the angle...
- and this is the terminal side.
- so this is a positive angele- "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 co-ordinates (a,b)
- the X value (where it intersects) is "a", the Y value (where it intersects) is "b"
- and I'm also, the whole point of what I'm doing here is,
- I want 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" (right-angled triangle)
- so to make it part of a "right-triangle" let me draw up an altitude...
- right over here, and let me make it clear that-
- this is a 90 degree agle
- 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 one
- so the the hypotenuse has length one!
- now what is the length of this blue side right over here?
- this blue side which you could view as the opposite side to the angle...
- well this height is the exact same thing as the Y co-ordinate of this point of intersection.
- so this height right over here is going to be equal to "b"
- (the Y co-ordinate 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 co-ordinate of this point of intersection
- and if you were to drop this down... this is the point: X equals "a"
- or this whole length between the origin and that, is of length "a".
- now that we have set that up-
- What is the co-sine of my angle going to be? (in terms of a's and b's and any other number 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're actually in the process of extending it)
- ..."SOH CAH TOA" definition of trig-functions...
- and the "CAH" part is what helps us with co-sine; it tells us...
- that the co-sine 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 one...
- so the co-sine of "theta" is just equal to "a"
- ...let me write this down again...
- so the co-sine of "theta"... is just equal to "a"...
- it's equal to the X co-ordinate of where the terminal side of this angel 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 one...
- so our sine of "theta" is equal to "b"
- so an interesting thing! This point where our terminal side of our angle intersected the unit circle at point (a,b)
- we can also view this, as...
- "a" is the same thing as co-sine 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 zero degree's (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 zero, or 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 just 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"
- or even larger- but I can never quite get to 90 degrees!
- at 90 degrees it's not clear that I have a "right-triangle" anymore, 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 consistant with "SOH CAH TOA"
- instead of defining co-sine as "oh if I have a right-triangle, which is the adjacent over the hypotenuse"
- and saying "sine is the opposite over the hypotenuse, tangent is the opposite over the adjacent"
- why don't I just say: for any angle, I can draw it in the unit circle using this convention I've just set up.
- and say that: the co-sine of our angle is equal to the X co-ordinate where we intersect.
- or where the terminal side of our angle intersects the unit-circle.
- I'll write... Where the terminal side of the angle intersects the unit-circle...
- and why don't we define sine of "theta" to be equal to the Y co-ordinate
- where the terminal side intersects the unit circle.
- so essentially for any angle- this point is going to define co-sine 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 co-sine of theta"
- which in this case is just going to be the Y co-ordinate, where we intersect the unit circle, over the X co-ordinate.
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