# Graphing Sine & Cosine w/out a Calculator Pt 2

Ok, we have y equals the sine of 2x minus
pi. Now this is how I am going to have my kids do this so hopefully you appreciate this
and it works for you. We are going to list off all the information that we know about
this graph then make a t table and sketch it. Ok, the amplitude is set by “a” in the
beginning of your equation and mine is equal to one so the amplitude is one. The amplitude
it the absolute value of a. The period of sine, cosine, secant, and cosecant is 2pi
over divided by “b” which is in front of the x. It is 2pi divided by 2 so the period is
pi. The phase shift, which I am going to just write ps, the phase shift is your left or
right horizontal movement, is c divided by b so it is going to be pi over two. Because
we have a minus sign in our parenthesis which is part of our standard form, a*trigfunction(bx-c)
+ d. That minus sign is there in the original formula so the phase shift is the movement
to the right. The graph is going to be moved pi over two units to the right. Our vertical
shift, which I am going to write as VS, our vertical shift is from a plus or minus of
a constant at the end of our equation and there is none. Our vertical shift is zero.
Now if we are doing this without a calculator, we need to know what we are counting by, that
is going to be one quarter of the period. So we are going to count by one quarter of
the period of pi, so we get pi over four. I want to minimize the amount of mental math
I have to do, I am going to write my phase shift, my count value, and my c value all
with matching denominators… common denominators to alleviate the mental math. We have pi over
four, pi over two, and pi. The common denominator between all of these values is four. I am
going to write this phase shift as 2pi over four and when I write this equation in the
t table I am going to write c as 4p over four. Again count by, phase shift, and c should
all be written with a common denominator to make the mental math a little bit easier.
Here is my t table. X… and Y equals sine of 2x minus 4pi over 4. As long as you know
how to deal with fractions and you know just the quadrantal angles of the unit circle,
the thinking parts of this problem are done. We are going to start at the phase shift which
is 2p over four. Don’t bother reducing this, that is why/how I want to do all this mental
math easily. Now I want to count by one quarter of the period so two over four plus one over
four is three over four, so 3pi over 4, 4pi over 4, 5pi over 4, and I need five points
so 6pi over 4. Again we are counting by one quarter of the period. Now I am going to show
all the steps in one of these lines but after that I am going to do mental math so I can
get through two examples in fifteen minutes or at least attempt to. The sine of two times
2pi over four minus 4pi over four. Well, two times two is four so that becomes the sine
of 4pi over four minus 4pi over four. That comes out to be zero and as you are doing
this work, if you start at the phase shift and count by one quarter of the period, if
pi over two, pi, three pi over two and back to two pi. If you do not get a trig function
giving you an answer of one, zero, or negative one you have done something wrong. So check
your work. That is zero and the sine of zero is zero. Now I am just going to verbally say
this. The sine of 2 times 3p/4 minus 4pi/4. Two times three is six over four. So six over
four minus four over four, six minus four is two and two over four is one half. So this
is the sine of pi/2 and the sine of pi over two is one. Plug in 4p/4. The sine of 2 times
4pi/4 minus 4p/4. That is eight and eight minus four is four over our common denominator
of four. Four pi divided by four is pi and the sine of pi is zero, so we are back to
0. The sine of 2 times 5pi/4 minus 4pi/4. Two times five is ten and ten minus four is
six over the common denominator of four. Six over four reduces down to three over two,
so this is the sine of 3pi/2 which is negative one. When you plug in the last one you are
going to get a value of zero. Here we have our t table set up and we are going to graph
two periods of this graph. Lay down the x axis which again is representing theta. We
have the values of 2pi/4, 3pi/4, 4pi/4, 5pi/4, and 6pi/4. Now those are the values that came
off my t table. They are all positive and I don’t know where zero is so I am going to
count backwards until I get to zero or I skip over it to a negative number. So 3, 2, 1,
ok 0, negative pi/4, negative 2p/4. Now that I know where zero is I am going to place my
y axis down here and mark off the values of one and negative one. Now I am going to start
graphing my sine function. Pi over two zero. Three pi over four and one. Pi, because four
divided by four is one, so pi zero. Five pi over four we are at negative one and at six
pi over four we are back to zero. Now here is what I have so far. Now do you remember
that the regular sine graph I did in the last video? The regular y equals the sine of x,
the parent function. It’s oscillation at zero zero. Well why is this one starting at 2pi
over four. Well it is because there is a phase shift of pi over two and 2pi/4 reduces down
to pi over two. That has pushed the graph to the right a little bit. What is the pattern
now? What if I wanted to graph two periods and not just one which is what I have got
here. Zero, negative one, zero, one, zero, and I have got these values on the x axis
to match my t table so the pattern is very easy now. 0,-1,0,1,0, ok so this point is
at pi over 4 negative one, back to zero, back to one, back to zero and then you see the
exact same pattern re-stamped basically from what I had over here. There is two periods
of y equals sine of two x minus pi. WHOOO!!! Let’s do another one. I think I have time.
I will just go until my fifteen minutes are up or do whatever I can manage to get done.
One more graphing problem. We have got y equals negative three cosine of one half x plus four.
No phase shift but we have a fractional b value which will increase the period. Amplitude
is the absolute value of a so it is going to be three. The period for sine, cosine,
secant, and cosecant is 2pi divided by b and when you pigtail that up, when you do 2 pi
divided one half that is the same as 2pi times two which is equal to 4pi. So since my b value
is less than one, it is actually going to stretch out the period and instead of oscillating
between zero and 2pi, it is going to take 4pi steps on the x axis to complete that period.
What else do we have. The phase shift, there is no plus or minus in my parenthesis so this
is zero. My vertical shift is my constant at the end, not inside my the math function
so it is four. We are going to count by a quarter of the period. What is the period?
The period is 4pi so we are going to count by one quarter of 4pi which is pi. Again,
if we are counting by a quarter of the period we are going to get those quadrantal angles
around the unit circle and get the easy values of zero, one, and negative one. T table. We
have x and y equals negative three cosine of one half x plus four. Books generally don’t
use parenthesis around this but I like to. This negative in front of the three is going
to reflect the graph over the cosine function over x axis. So instead of the function starting
at one it is going to start at negative one… maybe. Let’s see what the three does. We are
going to start our t table at the phase shift and count by a quarter of the period. And
we do not need to worry about common denominators because our denominator is one. We are starting
at the phase shift, which you are going to hear me say in all these videos now, and count
by a quarter of the period. So, pi, 2pi, 3pi, and we need five points so 4pi. Let’s plug
each one of these in. How much time do I have by the way? I better start moving! One half
of zero is zero and the cosine of zero, that is the x coordinate so that is one. So one
times negative three is negative three plus four. Negative three plus four is one. Plug
in pi. One half of pi is pi over two and the cosine of pi over two is zero, times negative
three is zero plus four which is four. Now are paying attention to what I am doing here?
I am plugging these in, I am working out, I am working my way back through the equation
until I get to leading coefficient and then I worry about the plus or minus at the end.
So 2pi times one half is 1pi. The cosine of ONE pi is negative one. This is negative one
here at 2pi and that times negative three is positive three plus four is seven. Let’s
plug in 3pi. The cosine of 3pi over 2 is, one, two, three pi over two, the cosine of
3p/2 is zero times negative three is zero plus four is back to four. Don’t…. Get the
five points before you start guessing what you think the pattern is by the way. And 4pi
times one half is 2pi. So the cosine of 2pi is again one. So the cosine of 2pi is one,
one times negative three is negative three and then plus four is back to one. So I have
all my points now and thankfully I have not run out of time. I have an x value of zero
so I don’t have to worry about where the y axis crosses. So we have zero, pi, 2pi, 3pi,
4pi, and this will give me one period. If I want I to go backwards or forwards I could
go 5pi, 6pi, 7pi, 8pi and so on and just start plotting my points. I don’t have to think
about the amplitude, the flip, the period change, or the vertical shift. All that is
just going to happen. We have zero one…we need a y value of 7 so 1,2,3,4,5,6,7. So zero
one, pi four, 2pi seven, 3p at four, then we are back down to one and that is one period
of cosine. And now I could do another pattern, another period and make if I like a second
period. You will probably be asked to graph two periods of these trig functions. Also,
in the minute that I have left here now that we have two periods of cosine. Instead of
starting at one, we sort of started at the bottom of the pattern, and in this case it
is not negative one, but it is at the bottom of the cycle. The amplitude. The overall height
is six from the top to bottom and if you take half of that you get the actual amplitude
of three. And by the way, as well, if you average these numbers of seven and one, if
you find the average of those numbers and get four. That happens to match the vertical
shift. So, if are ever given a graph and asked to find the vertical shift you average the
maximum and minimum values to find that vertical shift. ALL RIGHT. I am Mr. Tarrou. Thank you
for watching. GO DO YOUR HOMEWORK!!!

## 100 thoughts to “Graphing Sine & Cosine w/out a Calculator Pt 2”

1. ss98 says:

Amazing video as usual. You are my go-to math help. I cannot thank you enough! You are an absolute life saver. Thank you so much.

2. FunnyProductions24 says:

Gave me a greater understanding right in time for my test tomorrow. Thank you!!!

thanks so much for these videos. My professor didn't show us any kind of formula nor explain very well. I thought I could teach myself from our textbook but even the textbook makes it a thousand times harder than it is!

4. DTcrafts14 says:

When would you know how to flip the graph? Because on the last problem you did, you didn't flip it. Or would it always be flipped when you graph it?

5. Bekah J says:

You always pause and look over to your left, when you need to remember what to write.
I'm curious. Do you have an overhead over there or something? Lol.

6. joe figueroa says:

When you found the average at the of the video why is that you divided by 2 to get the vertical shift??…or is it that you just simply divide by 2 everytime to get the vertical shift.

7. Raul Torres says:

You just saved my life. I wish you were my professor. You have no idea how much I would like to thank you. Thank you for teaching.

8. Brenda Any says:

9. Alex Ho says:

THANK YOU SO MUCH

10. Shir Helmandi says:

why did you put 4pie over 4 in the table instead of 2pie over 4 ?

11. IphoneMaster says:

Wow I'am only year 9 and that looks really confusing I hope I get a better understanding when I'am older with this

12. Yeison Mazariegos says:

Thx your examples really helped a lot

13. NoA Naga says:

I love your enthusiasm when you teach!!

14. NBAPwns13 says:

When i put x into the equation i get decimals. I got like 3.9 , 3.8 , 3.7 …. for my 5 points

15. Dora says:

Ahhh!! you are magic. I have spent so much time stuck on this stuff and then in the first 2 minutes of this video you fixed all my problems. Thank you so much!

16. Bruno Marcelo says:

I LOVE U!

17. Karthik Mantha says:

Hey I just wanted to say that the graph of the second function is wrong (I think) it said cosine and you graphed it like  a sine function

18. kiyah carter says:

OMG this helped me understand so much!! This was way better than the Khan Academy video (which was not helpful at all).  Thank you!

19. Michael Richardson says:

The alarm in the background haha! These videos are amazing! They are actually teaching me unlike my teacher. Thanks a bunch!

20. tina duong says:

you saved my precal life

21. ProfRobBob says:

Only 511 more videos to Close Caption!!!

22. rualright96 says:

God, i love this teacher.

23. Jeremy Cutler says:

Thank you #ProfRobBob for the time you spend creating videos (and replying to comments). Your channel is like the 'porridge that was just right' from Goldilocks and the Three Bears (in comparison to a few other math help YouTube channels I've seen).

24. Scieneering says:

Thanks for your time on these videos and a huge thanks for taking the time to answer questions! I do have one quick question i'm a little confused on what the period is, i see tho that you keep using 2pie as the period. do i always use that? if not how do i determine what the period is?

25. Ryna says:

Thank you for sharing your knowledge and allotting time to teach us! 🙂

26. Matthew Hill says:

Your instruction REALLY cleared things up for me.  THANK YOU!!!  You are a gentleman and a scholar.

27. Dean Walker says:

Subscribed. Thanks.

28. Kiana Poier says:

Absolute lifesaver! You totally cleared this up for me. I love using the table. It's so helpful like BAM! Thank you!

29. Samantha Denning says:

Ahh you are awesome RobBob! Math is not my strong suit, I am a pre-med student and have to go quite a ways in math and you have helped me so much in understanding what the heck is going on! I feel like it's so simple now! Thank you for taking the time to make these videos, you are inspiring many other students other than the ones you teach personally I'm sure! Keep it up! ~Samantha
P.S. I love the enthusiasm, it made me smile big and loving what you do shows and makes a world of difference in our learning experiences! So thank you for that as well. Now, I have to go do my homework 😛

30. Maria Amoranto says:

thank you very much!! I can now catch up to our lessons…

31. Daniela says:

This definitely is helping me understand pre cal in the 10th grade 🙂 Thanks!!

32. Anthony Michalos says:

I am having trouble figure out what to count by with the problem of f(x)=2sin(x(pi/3))+1. I have got to be doing something wrong but cannot figure out what it is. If my period is 2pi and I divide by four I get pi/2, and the phase shift is pi/3 what do I count by? pi/6? If I do that I do not seem to get my key points from the unit circle… Anyone have any suggestions?

33. Vinh Nguyen says:

wow this actually helped, this guy explains much better than my monotone professor by far.

34. Laura Oleson says:

35. Leo G says:

Thank you for taking your time to make these videos, ProfRobBob. The only thing I'm having trouble understanding was when you set up your t-chart for the first example. Why did you change sin( 2x – pi) to sin( 2x – 4pi/4)? Can you explain where the 4pi/4 came from?

36. Karla García says:

Guess who's going to pass Trigonometry course thanks to #ProfRobBob  . This totally helped me out! My final exam is tomorrow and I'm feeling ready. THANKS A LOT 🙂

37. Liz Spears says:

What happens when you don't have a "pi" in your period?  y= -sin pi x .  amp is 1, period is 2pi/pi, PS 0 VS 0, count is 1/2.

38. Spencer says:

You are the man!

39. Epsilon says:

where did 4pi/4 come from in sinx = (2x-4pi/4)

40. John Tran says:

I'm sooo glad that I found you, but I wished I had found you earlier. Anyways you're a math lifesaver, keep up the great work!

Thank you so much. Phase shifts are hardest for me. Greetings from Finland! I have final exam tomorrow :))

42. Eduardo Garcia says:

43. Justin Williams says:

Do you have any videos like this with vertical & phase shift? Awesome video by the way!

44. Flyer Aerospace says:

You are the best professor in the world. Thank you very much for such an awesome lecture… CRYSTAL CLEAR 🙂

45. Sarah Kim says:

Thank You so much for making this video, I never really understood phase shifts but this helped me so much. I have a test tomorrow and I know I can get a good grade now.

46. Some Dude says:

Your videos and Khan academy are the only reason I am learning any trig while I'm in college.

47. Logan Phillips says:

I may be wrong but isn't pi the phase shift? When you do C/B doesn't this give the horizontal shift, not the phase shift?

48. john arellano says:

hello prof rob you never explained how yo get the count for x

49. Mary Belle says:

Thank you so much! (:

50. Dean Alvarez says:

like if Barclay sent you

51. L0k1 says:

This is amazing, thanks for explaining it so well, especially during the T-Chart.  My teacher explained it in class, but I didn't really understand too well, but this cleared it up so much!Thanks again!   🙂

you're my new hero, thanks a lot.

53. ZDson3112 says:

Thanks a million Professor Tarrou!!

54. TheseRightHereBitch says:

Got a 96% on my exam, thanks Rob! Earned a sub and like lol 😀

55. Travis Pellerin says:

you my friend saved me 500\$ in summerschool

56. Ahbab Murtoza says:

Thanks a bunch from the bottom of my heart professor Rob Bob for the outstanding videos! Your videos literally help me learn the concepts way better than text book and lecture classes, no joke. I do like your catchy word Bam!

57. Heather Leisure says:

Whoa yea now I'm all amped up for my trig homework

58. h says:

My teachers encourage me to use the calculator for everything. Luckily I never listen to her but instead seek the long way of doing things to gather a better understanding. Other students are so dependent on formulas and calculators that they have forgotten how to add fractions, crazy right.

59. Isaac Lall says:

so we always count by 1/4 of the period, like always?

60. Mark Landau says:

COMPILETELY AWESOME once again

61. Glitter Pink says:

These videos helped more than anything in the world!!! Thankful for you!!!

62. dustyandrea says:

My fellow students are getting crushed in our Trig class, and I am excited to share these videos with them, as they have helped me understand the material tremendously well. Thanks, and keep up the great work!!

63. Tayeb Elshaikh says:

8:09 I wasn't expecting that. Almost made me choke on a Dorito!

64. Kevin D says:

I have a better understanding thanks to your videos, thanks!

Thank you so much, Mr. Tarrou for your videos. I've learned a lot from them. Your style of teaching works for me. I was so lost in class especially regarding graphing sine, cosine and tangent. BAM! 🙂

66. Dylan Lander says:

Yeah Baby! We going to the top!! Titans of industry!

67. Steven Stratvert says:

Awesome videos! I am a musician trying to use math to create waveforms. Your videos have helped a lot. I have a question… What's the difference between sin(x+pi) and sin(x*pi)

68. Lauryn Frias says:

I thought that whenever you have y=sin(2x-π) you have to factor out the 2

Just out of curiosity, where do you teach Mr. Tarrou? (Great video btw)

70. Stipe Jurkovic says:

why do you do the easiest examples. just wasted my time.

71. John Wilson says:

awesome lecture, just watching your video on graphing over and over until your method sticks for my trig exam 🙂 thanks for doing what you do

72. Just Another Name says:

does the t table always work? I used the t table and compared with my answer and it didn't math

73. Brianna Limon says:

Thank you so much sir. You're teaching is amazing and calming.

74. Jess H says:

so much better than my math teacher omg

75. Sidney Goodrich says:

6:55 "4 divided by 4 is pi" XD I know what you meant but it just sounded funny. hahaha

76. David Bowling says:

You didn't even jump into the video…. thumbs down.

77. Aly says:

thank you so much, this is so helpful
x

78. Matthew Suksnguan says:

Mr. Tarrou would 0<bx-c<2π work as well?

79. faith murray says:

Why would it be 4pi over four I do not understand.

80. Jillian Carrell says:

Why isn't the phase shift just pi instead of pi/2

Please explain to me why you arbitrarily create the count. When I graph y=sinx on a calculator, the line goes through pi and 2pi. You seem to arbitrarily divide the period by 4 to create a totally different  graph.

82. jbgoogl says:

Thanks! You earned a subscriber. 🙂

83. Josheb E. Mangru says:

Thank you very much ProfRobBob. You are a blessing.

thank you so much for these videos and your explanations, you have made my math class much easier.

Would the count 1/4 work for all sine and cosine equations? Such as y=(x+pi/3)

86. hellob.l96 says:

Thank you so much. I have an exam tomorrow and I’ve been so confused in class even after I ask questions

87. Sophia says:

Thank you so much!!!!!!!!!!

88. Rowan Irons says:

You're the best teacher I've ever had thank you so much

89. Nawal Amir says:

Is the P.S. 0 at 9:30 because there is no c in c/b, so 0/(1/2) is 0

90. Rodrigo Gomez says:

You explain concepts very well. Thanks for your time, I appreciate it.

91. Fuck Isis says:

I am a math Teacher and i admit it, u explain it better than me.

92. Esha Angel says:

Saved me for my Alg2/Trig test

93. bbs boss gaming GAMING says:

Really helped me, thx fopr the amzing video!!!

94. Chloe Zhou says:

I still don't understand why the ps is pai/2, not –pai

95. fatima al ali says:

This is so useful

96. Sharon Myers says:

Fast and complete! You, sir, are amazing! Thank you for your efforts. This has helped me immensely.

THANKs….amazing wohooo

98. Ghost World says:

It's bizarre and a testiment to your teaching that after all this time, these are still some of the best vids (you and Professor Leonard need to get together and do a collaboration series of videos!) What still gets me is that the Phase Shift is sort of "backwards" a positive phase shifts it to the left and a negative Phase shift pushes it right…which looks 'wrong'! My question though, is that do we always count by 1/4 of the period for the X axis?