BAM!!! Mr. Tarrou
If you are looking for a video of how to graph sine and cosine functions with all your changes in
amplitude, phase shift, and vertical shifts, this is not the video you want to watch.
I just want to tie..that will be the next video I post... I just want to tie in how you get the numbers off the unit circle and how that translated into a graph on the x y axis
So we have a unit circle drawn out. Handy dandy. My students are getting drilled constantly about these unit circles.
The cosine of theta is x over r and sine of theta is y over r.
Since this is a unit circle, all the x coordinates therefore are cosine and all the y coordinates around this unit circle are sine values.
Now, we are going to graph sine and cosine hopefully in the next 15 minutes.
Just basic trig functions. No transformations.
off the values we get from this unit circle. When we graph these trig functions plotting that many points we are going to need the help of our calculator.
I went ahead and calculated the value of the square root of 3 over 2 and got .87
Now after this lesson I will show you how I am going to teach my own students how to graph trig functions, they are not going to be allowed to use calculators, so something is going to need to change from plotting all these points.
So we have y equals the sine of x. We are going to try and see what that looks graphically.
Now I am going to say y over here, but on a unit circle, which is on the x y axis...this is the origin, the x coordinates are cosine and the y's are sine.
But, what you do as a graph of Sine and Cosine is you track the values of sine, or cosine, or tangent...whatever you are doing... but we are only doing sine and cosine today, we track the change in those values over the x axis.
...almost like over time. As we rotate what do those actual values become.
Don't forget what do you put into a trig function?
You put in an angle measure and I am going to do all my angles in radians.
So, the x axis is actually the values of theta. The x axis is the domain and we plug in angle measures
to find, at least as far as trig functions are concerned, the ratio of the sides of a right triangle. In terms of Sine this is y over r.
Lets see if we can't plot a bunch of points and see what these basic sine and cosine functions look like.
Zero degrees. The sine of 0 degrees, the x axis being the domain of theta, the sine of 0 degrees is 0.
So, we have the sin(0) equals 0.
At pi/6 we have got this vertical distance because sine is y over r.
So when you rotate pi/6 radians, the y value is 1/2 and I don't need a calculator for that. So at pi/6 the sine value is 1/2.
At lets say pi/3,
Well if you rotate pi/3 radians, the y value is or the sine value is square root of 3 over 2 which we know is .87...or at least I did after I used my calculator.
So at pi/3 we are at almost .9. I am basically in the middle, I am doing more than just a sketch but I am using chalk and chalk board so .87 or .9 you cannot tell the difference visually.
At pi over 2. Let's pick a different color here. At pi/2, at 90 degrees, the y value is what?
When you rotate pi/2 radians, the y value ... the y coordinate is 1. So BOOM, there we go.
If there has been no transformations like if there is no amplitude change which I will speak about in the next video, the highest value you ever get is one
because the y value and the radius is the same. This is because you are moving perfectly vertically, up and down... perfectly vertically:/
Anyway the radius is 1, the y value is 1 thus you get the largest value sine can take on without an amplitude change.
Now as we rotate back over here to 2pi over 3, we are starting to come back down the circle so the y values are going to start to reduce or get smaller.
So, the sine of 2 pi over 3 is once again square root of 3 over 2, or basically .9.
Now why again does pi/3 and 2pi/3 have the same basic coordinates except for a sign change?
Because they both have the same reference angle of pi/3 radians, or in degrees 60 degrees.
You might notice also that I skipped over pi/4.
When you count or set up an x axis, or any axis on a graph, you cannot change by what you count.
So I can't count by 30 degrees then by 15 and 15 and then 30 again.
You can see I already have a crowded x-axis and I certainly do not want to count by half or 15 degrees which would be pi/12.
I am just going to count by 30's or pi/6 as we go around the unit circle since that is how I set up my x axis which is representing theta, the domain value.
As we come back over here to 5pi/6, the y value is again 1/2.
At pi the sine value is 0.
Now these values are going to seem to be repeating, but they are all going to be negative because we are falling below the x axis which means your y values are going to be negative of course.
At 7pi/6 we get -1/2.
At 3pi/4 we get negative square root of 3 over 2 which is basically negative .9 or -.87.
At 3pi/2 we are once again down at the bottom where we have a quadrantal angle.
We did a quadrantal angle and the sine value, the y value down there, at 3 pi/2 the sine of 3pi/2 is -1
That is going to be the lowest value that sine can take on unless there is a stretch, unless there is an amplitude change, because the y value and the radius are equal.
So we get -1
Coming into quadrant 4 our y values start to rise again.
So we have -.87 and then -1/2 and then we are finally back to 0.
So my sine graph looks like this.
Now, if I keep the pattern going on, hopefully you catch on to that and if not just ask me a question.
I didn't hear any:)
Again 30 degrees and I am going to highlight those angles that have the same reference angles so they all have similar coordinates.
If I keep this pattern up, we have all the colors so I can speed this up, one tick over and I am at -.5 instead of .5. One tick over and now I am at -9.
Another tick over and I am at negative one.
Now I am going back up to -.87
Another tick over and we are at -.5.
And then at 0 again.
Another tick over and I am at uhh
Let me see, going back to zero and now a tick to the left I am at .5
And now at .9 essentially, 1, now .87, .5, and at finally at 0.
Now if I connect all these dots I get that picture.
By the way, these are negative angle rotations ...
negative...I didn't put all the negative values, I skipped some labels.
Negative pi/6 gives -.5
-pi/3 is -.87
-pi/2. Now this says 3pi/2 but those are coterminal angles. -pi/2 radians gives -1
Look what we have, we have a nice smooth wave.
This could be a sound wave, or wave in the ocean,
electrical waves, light waves, it is a sine graph.
And you might notice that with this graph, you can rotate it 180 degrees and it will be a perfect match.
It would line up exactly.
If you can take a graph and rotate it 180 degrees, that means that this is an odd function.
Lets take a look if we have time at Cosine.
I am going to do this a lot quicker, so I hope you understood the previous example.
Cosine Well y equals cosine of x.
We are still going to plug in angle measures
but when we evaluate cosine
we are not looking for the y values around the circle we're looking for the x values around the circle.
So the cosine of 0 degrees is 1
Lets use a completely different color.
Not sure if the yellow and the white are going to highlight that much of a difference.
So the cosine(0) is 1 and the cosine(pi/6) is square root of 3 over 2. We heard that number before!
Square 3 over 2 is .87, so (pi/6,.87)
The cosine(pi/3) is 1/2.
At pi/2 our cosine is 0.
At 2pi/3, the cosine of 2pi/3 is 1/2.
Hopefully as I draw this you can see the pattern showing up on our graph.
At 5pi/6 our cosine value is negative square root of 3 over 2, or approximately -.87.
And now at pi, we are now to the farthest to the left on the circle as we will get.
The x value will be the smallest or most negative it will get, and that is -1
And the radius is 1, so the cosine of pi is negative 1.
Ok, if I can stop pointing to the unit circle...well I will finish it up.
Then we start in quadrant 3 and 4
having the x values become larger or more positive.
So, at 7pi/6 we are at negative square root of 3 over 2.
or basically -.87
At 4pi/3, 1...2...3...4pi/3, the cosine is -1/2
At 3pi/2 our x value is 0.
At 5pi/3 we are 1/2.
And then finally we are back at .87
...and then at 1.
If I keep that pattern going on the other side. Lets see the graph so far.
Here is y equals the cosine of x.
I probably should have that in blue and not yellow.
Now if I keep that pattern up: 1, .87, .5, 0,
negative point 5, -.87, -1
I am going this way now, you can see my negative angles wrapping clockwise around the unit circle.
- 7pi/6 we are at negative square root of 3 over 2.
then at -.5...0...positive 1/2...positive .87
and now at 1.
So again, -pi/6 positive square root 3 over 2
Negative pi/3 positive 1/2
-pi/2, 0
-2p/3 we are at -1/2
Starting to see the connection between the unit circle and the graph I hope.
So we get this graph.
That is y equals the cosine of x.
Now do you see the symmetry that we have on the y axis?
If you have a graph that is symmetric to the y axis, that means that it is an even function.
That might come in handy with problems we will do later in Trig.
So, this is how we track rotation along the x axis.
and pull those numbers off the unit circle.
Now in the next lesson you are going to see me...
We don't want to plot this many points.
We want to make this as quick and easy as possible and with my students am not going to allow them to use a calculator.
What I am going to show them to do, and you on the video
is show you how to just find the easy values of 0, 1, 0, -1, and 0 again.
We are in the next lesson key in on just those easy points.
When you are finding sine and cosine values of 1, 0, and -1, now we are just talking about the quadrantal angles.
We will talk about a system to set up t-tables that will only allow us to get these quadrantal angles.
Thus will do these graphs without the aid of a calculator and without plotting so many flipping points!!!
WOOOO!!! I am Mr. Tarrou, go do your homework:D