Question

find f(θ)=cos(θ)

Answer 1

find \[f(\theta)=\cos(\theta)\]
amplitude
period
domain
range
x-intercept

Answer 2

Do you know how the function looks on a graph?

Answer 3

no

Answer 4

It's pretty hard to know the characteristics of cos without the graph so let me find a picture

Answer 5

ok thank you

Answer 6

This is how it looks like.

Answer 7

Ok. The amplitude of a cyclic function (ie. cos and sin) is basically
(max value of the function -min value of a function)/2

Answer 8

So what would that be?

Answer 9

Let me be pedantic about the terminology. This is a periodic function and not a cyclic function, my bad.

Answer 10

max value =1
min value=-1
1-(-1)/2=1 is it right?

Answer 11

Yes

Answer 12

Now let's move on to finding the period.

Answer 13

Notice that cos repeats itself on the graph.

Answer 14

See that cos is the same between -2*Pi to 0 and 0 to 2*Pi

Answer 15

ok?

Answer 16

The period is the length of each cycle.

Answer 17

Looking at the graph, -2*Pi to 0 is one cycle and 0 to 2*Pi is another cycle.

Answer 18

so the period would be 2?

Answer 19

What is the length of each cycle?

Answer 20

2?

Answer 21

Close... do you see \[2\Pi\] on the x-axis?

Answer 22

yes

Answer 23

The length would be \[2\pi - 0\] or \[0 - (-2\pi)\]

Answer 24

2pi

Answer 25

??

Answer 26

Yup.

Answer 27

Do you see now that the period is the length of one cycle?

Answer 28

yes thanks

Answer 29

The domain of a function is the set of values where it is possible to get a value when you enter it into the function. The range of a function is the set of values that you can get from the function.

Answer 30

i think the domain is \[(-\infty, \infty)\]

Answer 31

What would be the range?

Answer 32

the same?

Answer 33

or (-1,1)??

Answer 34

Remember that the range is the set of possible values that you can get from the function. Looking at the graph, all the values are within 1 and -1

Answer 35

and including 1 and -1.

Answer 36

The second answer is close.

Answer 37

so it's not (-1,1) what is then?

Answer 38

Did you learn "[" and "]"?

Answer 39

They mean that the numbers are included in the interval

Answer 40

i don't think so

Answer 41

(-1,1) is equal to saying
\[-1<x<-1\]

and
[-1,1] is equal to saying
\[-1 \le x \le -1\]

Answer 42

I guess you get to learn this later.

Answer 43

Thre range is [-1,1]

Answer 44

ohhh okay now all i need is the x intercept!

Answer 45

To find the x intercepts, set the function to 0. \[0 = \cos \theta\]Then you find what values of \[\theta \]makes the function 0. You can look at the graph or calculate it algebraically.

Answer 46

how?

Answer 47

Actually to calculate it algebraically, we need to know the x intercepts of the general function which is the one we are doing right now.

Answer 48

ok so how do i find the x intercepts?

Answer 49

Looking at the graph, y is equal to 0 when theta \[... , \pi/2, 3\pi/2, ...\]

Answer 50

so the x intercepts would be \[\Pi/2, 3\Pi/2 ???\]

Answer 51

Yup. Notice there are multiple repeating x-intercepts.

Answer 52

So we have to write it as \[\pi/2 + n2\pi\] where n is all integers

Answer 53

Notice \[2\pi\]in the answer. That is our period.

Answer 54

ok thank you so much u were an awesome teacher not only i did my work but i was able to understand it! thank you :)

Answer 55

Here to help. Merry Christmas

Answer 56

merry christmas