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## anonymous one year ago Describe the graph of the cosine function.

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1. anonymous

@ganeshie8

2. anonymous

@nincompoop @Preetha

3. anonymous

@campbell_st

4. campbell_st

so do you know what the graph looks like...?

5. anonymous

Hehe, not really

6. anonymous

Let me check online

7. anonymous

Pretty similar to the sine graph if you ask me...

8. campbell_st

here is a site that will graph it for you on the left side just enter y = cos(x) and you'll see the graph https://www.desmos.com/calculator

9. anonymous

ok, so now, just to make sure, the domain is all real numbers, the range is -1≤y≤1, the y intercept is 1, but what would the x- intercepts be??

10. anonymous

@campbell_st

11. campbell_st

so where does it cut the y-axis, are you working in radians or degrees..?

12. anonymous

Degrees

13. anonymous

@campbell_st

14. anonymous

Sorry to bother u, but I'm pretty lost...

15. campbell_st

so so cos(90) =0 and cos(270) = 0 so the x- intercepts are at x = 90 and x = 270 then repeat every 180 degrees

16. anonymous

Oooh, ok, so, as an answer, I put "starting at 90°, every 180°"?

17. campbell_st

you could...

18. anonymous

would it be right?

19. anonymous

@ganeshie8 could u help me out pleease??

20. SolomonZelman

There is a pattern to x-intercepts. For what values is cos(x)=0, the solutions to this are the x-intercepts. For example at x=90 the cos(x) is 0 and so it is for x=270, for x=450 and on adding 180º each time.... $$\large\color{black}{ \displaystyle x=90^\circ }$$ is your first y-intercept then, $$\large\color{black}{ \displaystyle x=90^\circ+180^\circ=270^\circ }$$ is another x-intercept then, $$\large\color{black}{ \displaystyle x=90^\circ +180^\circ +180^\circ+.... }$$ Each time you add 180º you get an x-intercept. So, you can generate the pattern: $$\large\color{black}{ \displaystyle x=90^\circ +(180^\circ\times {\rm k})}$$ (for all positive integer values of k, and for 0 as well) You can also go -180º , subtract 180º, to get the x-intercept. $$\large\color{black}{ \displaystyle x=90^\circ-180^\circ=-90^\circ }$$ is an x-intercept $$\large\color{black}{ \displaystyle x=90^\circ-180^\circ-180^\circ=-270^\circ }$$ $$\large\color{black}{ \displaystyle x=90^\circ -180^\circ -180^\circ-180^\circ-.... }$$ So, you can generate the pattern: $$\large\color{black}{ \displaystyle x=90^\circ -(180^\circ\times {\rm k})}$$ (for all negative integer values of k) -------------------------------- it follows that x-intercepts all go by the pattern $$\large\color{black}{ \displaystyle x=90^\circ -(180^\circ\times {\rm k});~~~\color{blue}{\rm \forall~~k\in{\bf Z}}}$$

21. SolomonZelman

The blue notation here means "for all integer values of k"

22. SolomonZelman

saying that k can be equal to ..... $$-5$$, $$-4$$, $$-3$$, $$-2$$, $$-1$$, $$0$$, $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, .....

23. anonymous

Hehe, sorry @SolomonZelman , I'm not quite following... XD

24. SolomonZelman

Its alright....

25. anonymous

So what you're saying is that the x-intercepts are all multiples of 90?

26. anonymous

@SolomonZelman

27. SolomonZelman

the x-intercepts start from 90º. and they are all the following: x=90-180 x=90-180-180 x=90-180-180-180 x=90-180-180-180-180 x=90-180-180-180-180-180 x=90-180-180-180-180-180..... and so on.... x=90+(k•180) where K IS NEGATIVE integer or 0. And ALSO, x-intercepts are from 90 and +180 x=90+180 x=90+180+180 x=90+180+180+180 x=90+180+180+180+180 x=90+180+180+180+180+180 x=90+180+180+180+180+180...... and so on.... x=90+(k•180) where K IS POSITIVE integer or 0.

28. SolomonZelman

this why, you can generate a pattern with k (where k is both positive integers and negative integers and 0) and this pattern is: x=90+(k•180)

29. anonymous

Is 180 an x-intercept? No right?

30. anonymous

Aren't the x-intercepts the same as the sine and tangent graphs?

31. SolomonZelman

no 180 is not, but 90 plus 180 (add or subtract 180 from 90 any number of times)

32. SolomonZelman

lets look at it simple. (we are talking about the cos(x)=y graph) the x-intercepts are the values of x that make the y=0. these are as I posted: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ the x-intercepts start from 90º. and they are all the following: x=90-180 x=90-180-180 x=90-180-180-180 x=90-180-180-180-180 x=90-180-180-180-180-180 x=90-180-180-180-180-180..... and so on.... x=90+(k•180) where K IS NEGATIVE integer or 0. And ALSO, x-intercepts are from 90 and +180 x=90+180 x=90+180+180 x=90+180+180+180 x=90+180+180+180+180 x=90+180+180+180+180+180 x=90+180+180+180+180+180...... and so on.... x=90+(k•180) where K IS POSITIVE integer or 0. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

33. SolomonZelman

so they are ....... -450º, -270º, -90º, 90º, 270º, 450º, .... (adding or subtracting 180 each time)

34. anonymous

Yes, I understand now! So examples of intercepts would be 90, 270, 450, 630, 810, etc?

35. SolomonZelman

yes, or you can subtract 180 from 90 many times 90 90-180=-90 -90-180=-270 -270-180=-450 and on....

36. SolomonZelman

So, for this reason they can be all given using a pattern x = 90 + k•180 for all integers of k. (is this still unclear?)

37. SolomonZelman

(all integers = ..... −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, ..... )

38. anonymous

Great, and just to make sure, the rest of the cosine graph would be: y-intercept is 1, the domain are all real numbers and the range is -1≤x≤1

39. anonymous

right?

40. SolomonZelman

the rest of the cosine graphs?

41. anonymous

No, I mean the rest of the characteristics of the cosine graph

42. SolomonZelman

yes, yes. But provided they are not shifted sideways. they can have any angle of cx, such that y=cos(c•x)

43. SolomonZelman

and the range will change if you shift it up/down

44. SolomonZelman

so y=cos(x), has a range [-1,1] y-intercept 1

45. SolomonZelman

but, y=cos(x+c) has a range of [-1,1] y-intercept 1-c

46. SolomonZelman

and y=cos(x)+c has a range of [-1+c,1+c] y-intercept of 1

47. SolomonZelman

this way y=cos(x+a)+b has a range of [-1+b,1+b] y-intercept 1-a

48. anonymous

I just started learning the trig graphs, so I don't think the curves will shift. thanks anyways! One more thing, is it fair to say that the x-intercepts are all odd multiples of 90?

49. SolomonZelman

Yes, negative or positive multiples of 90

50. SolomonZelman

I mean negative or positive, odd multiples of 90

51. anonymous

Thank you soooo much!!!

52. SolomonZelman

in this case of y=cos(x), which is not the case if you shift the graph sideways such that y=cos(x+b), and which is not [neccessarily, but for most values in fact not] the case if you multiply the angle or the function times a scale factor, such that: y=cos(bx), or y=bcos(x).

53. SolomonZelman

this is just graph shifts..... the rule you can get in a book or online..... my PC is about to resart because I am scanning and updating. cu

54. anonymous

Thanks!

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