Question

Can you help me understand this assignment?

Answer 1

Choice #1: Describe each of the following properties of the graph of the Cosine Function, f() = cos() and relate the property to the unit circle definition of cosine.
•Amplitude
•Period
•Domain
•Range
•x-intercepts

Answer 2

\[f(\Theta)=\cos(\Theta)\]

Answer 3

@mathmale

Answer 4

Found you!
Sorry for the delay!
Compare f(x)=cos x to
g(x)=a*cos (bx + c). This is the most general form for the equation of the cosine function.
Here's what the letters represent:
a: amplitude
b: frequency
c: horizontal phase shift

Answer 5

If you compare your function to y=a*cos (bx+c), you'll see that a=1, b=1 and c=0.
Could you be comfortable with that?

Answer 6

Got it.

Answer 7

This is the first step in describing the function with amplitude, period, domain, range and x-intercepts.

We have a little more work to do to come up with the amplitude and period of y=cos x.

The AMPLITUDE is actually the ABSOLUTE VALUE of a. Here, a=1 and the absolute value of that is also 1.

The PERIOD is (2pi)/b. Since your b=1, the period of this cos function is (2pi)/1, or 2pi.

The DOMAIN is the set of all input values (x-values) that are permissible. We could discuss that further if need be. Here, since our focus is the cosine function, the domain consists of the set of ALL REAL NUMBERS. The sine function has the same domain.

Answer 8

The RANGE is the set of all POSSIBLE y-values. Since the largest y value your y=cos x can have is 1, and the smallest -1, your RANGE is simply [-1,1}: All real numbers between -1 and 1, inclusive.

Answer 9

Lastly, BG, the x-intercepts are the x values at which y=cos x=0. Since the cosine function is periodic, it repeats itself every 2pi radians. If you have some ref mat'l available, please look up "cosine" on the Internet.

Answer 10

Most articles you'll find there will give you a graph of the cosine function.
Note, from this graph, where the cosine = 0. Does this make sense?

Answer 11

You could use this outline as a reference helpful in identifying the same descriptors of other cosine functions or of sine functions where a, b and c are different from our current a=1, b=1 and c=0.

Answer 12

So, would the x-int's be (-3pi/2,0), -pi/2,0), (pi/2, 0), and (3pi/2, 0)???

Answer 13

And what does it mean when it says to relate them to the unit circle definition of cosine?

Answer 14

@mathmale

Answer 15

BG: I'm sorry for the delay.

Answer 16

You're fine. :)

Answer 17

You came up with a list of x-intercepts. Mind taking a quick look at that list and figuring out what the separation between the zeros is? In other words, how far is it between -pi/2 and +pi/2?

Answer 18

4pi? I don't know..

Answer 19

Interesting trying to relate this to the unit circle def'n of cosine. I'll get to that in a moment. Regarding my question for you: Try finding the distance between those 2 x-values by subtracting (-pi/2) from (+pi/2).

Answer 20

Think: What is 0.5-(-0.5)?

Answer 21

How are you doing, BG?

Answer 22

So it would just be pi?

Answer 23

Perfect. If you were to graph more than one cycle of the cosine function, e. g., from 0 to 6pi, you'd see that cos x=0 every pi distance along the x axis. Make sense to you? This is important.

Answer 24

Yes. :p

Answer 25

Were my x-int's right, or no?

Answer 26

OK! So, instead of listing -pi/2, pi/2, 3pi/2, and so on, you could describe the x-intercepts as pi/2 plus or minus n times pi. What this says is: if we use pi/2 as a starting point, the next x-intercept is pi units to the right, the next one 2pi units to the right, the next one 3 pi units to the right, and so on.

Answer 27

Yes, your x-intercept coordinates are perfect. They'd be appropriate if you were focusing solely on the interval [-2pi,2pi]. My suggestion is for the more general case, where each new x-intercept is an integer multiple of pi to the right of your starting point (any zero of y=cos x).

Answer 28

Awesome. :)

Answer 29

So, how far along are we in responding to this question of yours? I'd be glad to spend more time on it if need be, or you could post a new problem to consider.

Answer 30

I've got all the things on the list, I just need to know how to relate them to that unit circle def. of cosine thing. What does that mean?

Answer 31

@mathmale

Answer 32

I sketched such a circle, noting that one x-intercept is at pi/2. If I travel counterclockwise around the circle by increasing my angle by pi, I get 3pi/2. If I travel again in the same manner, I get 5pi/2. And so on. Just another (very convenient) way to predict where your x-intercepts will be.

Answer 33

This is what it tells me on my website, for how to do it:
"For example, if discussing the domain, describe how the domain of sine or cosine is based off of the unit circle."
But what does that mean?

Answer 34

I'll be back, but seeing that Campbell is responding, I'll let him/her finish.

Answer 35

here is a simple connection between the unit circle and cos