Graphing Sine and Cosine By Hand

[harpazo]

What is the easiest way to graph the functions y = sin x and y = cos x WITHOUT a calculator? I have not done this in years. I came across this topic in my self-study of Precalculus. I recall that a table of values must be set up. I also recall that the values selected for x must be in radian measure. Correct?

 

[Dr.Peterson]

It depends on how much accuracy you want. (It isn't really necessary to use radians, but the functions are properly defined in terms of radians, and some teachers would insist on it more than others, largely depending on context.)

I would start a sketch by plotting the quadrantal angles, where the value is 0, 1, or -1; then I'd fill in intermediate values as much as I chose to.

 

[firemath]

Something interesting....
Look at how the graphs of those two intersect (sin x in red and cos x in blue)

(I know it's not in radians; Desmos is screwy.)
You can see that they both deviate only 1 unit from the x-axis (You probably know that already.)
Both of the graphs are on alternating multiples of [MATH]\pi[/MATH]. Start your graph by marking those; it's one less step to mark them, and opposed to tabling them out.

 

[harpazo]

It depends on how much accuracy you want. (It isn't really necessary to use radians, but the functions are properly defined in terms of radians, and some teachers would insist on it more than others, largely depending on context.)

I would start a sketch by plotting the quadrantal angles, where the value is 0, 1, or -1; then I'd fill in intermediate values as much as I chose to.

1. Not looking for accuracy. Just a simple graph of each without a calculator. Using Desmos seems to be the favorite route by most teachers and students.

2. I am not a student. This is a personal review of Precalculus.

3. Most students have forgotten how to graph by hand.

 

[harpazo]

Something interesting....
Look at how the graphs of those two intersect (sin x in red and cos x in blue)
View attachment 19040
(I know it's not in radians; Desmos is screwy.)
You can see that they both deviate only 1 unit from the x-axis (You probably know that already.)
Both of the graphs are on alternating multiples of [MATH]\pi[/MATH]. Start your graph by marking those; it's one less step to mark them, and opposed to tabling them out.

Desmos is the easy road to take for most teachers and students today. Forgetting how to graph by hand catches up to students who are later instructed to graph 3D images in Calculus 3.

 

[Dr.Peterson]

Something interesting....
Look at how the graphs of those two intersect (sin x in red and cos x in blue)
View attachment 19040
(I know it's not in radians; Desmos is screwy.)

Actually, that is in radians! One half-cycle starts at 0 and ends at [MATH]\pi[/MATH], which is about 3.14.

Students often imagine that if it's in radians, it has to mention [MATH]\pi[/MATH], because that's how they always see it. But radians are just numbers. (And you can tell Desmos to measure either axis in terms of [MATH]\pi[/MATH] if you wish -- click Settings, then Step.)

 

[harpazo]

Actually, that is in radians! One half-cycle starts at 0 and ends at [MATH]\pi[/MATH], which is about 3.14.

Students often imagine that if it's in radians, it has to mention [MATH]\pi[/MATH], because that's how they always see it. But radians are just numbers. (And you can tell Desmos to measure either axis in terms of [MATH]\pi[/MATH] if you wish -- click Settings, then Step.)

Do you agree that it is important for students not to forget how to graph, at least, basic functions by hand?

 

[Dr.Peterson]

Do you agree that it is important for students not to forget how to graph, at least, basic functions by hand?

Certainly. We teach most concepts first "by hand" to promote understanding, before using tools to do the tedious parts when they are no longer what's important. If one forgets how to graph the basics, then one has forgotten what a graph means, and has become uneducated.

So, whether or not other students forget (I have no evidence one way or the other), it's important for you to learn it and keep it. (If you're trying to learn, you're a student, whether or not you have a teacher. And someone who isn't trying to learn isn't really a student, even if they do have a teacher.)

 

[harpazo]

Certainly. We teach most concepts first "by hand" to promote understanding, before using tools to do the tedious parts when they are no longer what's important. If one forgets how to graph the basics, then one has forgotten what a graph means, and has become uneducated.

So, whether or not other students forget (I have no evidence one way or the other), it's important for you to learn it and keep it. (If you're trying to learn, you're a student, whether or not you have a teacher. And someone who isn't trying to learn isn't really a student, even if they do have a teacher.)

I concur.

 

[Otis]

What is the easiest way to graph the functions y = sin x and y = cos x WITHOUT a calculator? I have not done this in years …

… Not looking for accuracy. Just a simple graph of each …

Use a reliable, labeled graph of each to memorize the shape, intercepts and turning points. That is, the easiest way to draw a rough sketch of sine or cosine is from memory. For people who struggle to remember, I'd suggest taping enlarged copies to the wall where they study trig (worked for me).



Note some basic patterns.

The x-intercepts are multiples of pi (…-4pi, -3pi, -2pi, -pi, 0, pi, 2pi, 3pi, 4pi…). Plot those points first.

The turning points occur halfway between the x-intercepts (…-2½pi, -1½pi, -½pi, ½pi, 1½pi, 2½pi…). Plot those points next, alternating between peaks and valleys. Use improper fractions, instead of mixed numbers, when labeling the x-axis (pi/2, 3pi/2, 5pi/2).

The graph of sine passes through the Origin (the graph of cosine does not). Near the Origin, the graph looks like y=x. Use these facts to begin your rough sketch, by drawing a line upwards through the origin with slope 1 and then curve to meet the first turning point -- the peak at (pi/2,1). The curve goes down through the first x-intercept to the right of zero (pi,0) with slope -1. Continue to the next turning point -- the valley at (3pi/2,-1), then back up to the second x-intercept (2pi,0). That is one period.

The graph of cosine is identical to the graph of sine shifted pi/2 units to the left.

?

 

[harpazo]

Use a reliable, labeled graph of each to memorize the shape, intercepts and turning points. That is, the easiest way to draw a rough sketch of sine or cosine is from memory. For people who struggle to remember, I'd suggest taping enlarged copies to the wall where they study trig (worked for me).

View attachment 19047

Note some basic patterns.

The x-intercepts are multiples of pi (…-4pi, -3pi, -2pi, -pi, 0, pi, 2pi, 3pi, 4pi…). Plot those points first.

The turning points occur halfway between the x-intercepts (…-2½pi, -1½pi, -½pi, ½pi, 1½pi, 2½pi…). Plot those points next, alternating between peaks and valleys. Use improper fractions, instead of mixed numbers, when labeling the x-axis (pi/2, 3pi/2, 5pi/2).

The graph of sine passes through the Origin (the graph of cosine does not). Near the Origin, the graph looks like y=x. Use these facts to begin your rough sketch, by drawing a line upwards through the origin with slope 1 and then curve to meet the first turning point -- the peak at (pi/2,1). The curve goes down through the first x-intercept to the right of zero (pi,0) with slope -1. Continue to the next turning point -- the valley at (3pi/2,-1), then back up to the second x-intercept (2pi,0). That is one period.

The graph of cosine is identical to the graph of sine shifted pi/2 units to the left.

?

How about making a table like the one below?

x.........sin x

0.........0

pi........0

etc........

Is this ok?

What about when the question is specific?

Example: Graph one cycle of y = sin x.

 

[Otis]

How about making a table like …
x.........sin x
0.........0
pi........0
Is [that] ok?

Yes, it's always okay to organize coordinates in a table, before graphing.

What about when the question is specific?
Example: Graph one cycle of y = sin x.

That's not specific because it doesn't specify where the cycle starts. In such a case, you're free to graph sine on any interval of length 2pi.

I would choose from x=0 to x=2pi because those x-intercepts and turning points are easy to remember:

(0, 0) \(\quad\) (pi/2, 1) \(\quad\) (pi, 0) \(\quad\) (3pi/2, -1) \(\quad\) (2pi, 0)

Connect the points with a smooth wave.

?

 

[Steven G]

Here is a link to the method I use to graph sin and cos functions.

 

[harpazo]

Yes, it's always okay to organize coordinates in a table, before graphing.


That's not specific because it doesn't specify where the cycle starts. In such a case, you're free to graph sine on any interval of length 2pi.

I would choose from x=0 to x=2pi because those x-intercepts and turning points are easy to remember:

(0, 0) \(\quad\) (pi/2, 1) \(\quad\) (pi, 0) \(\quad\) (3pi/2, -1) \(\quad\) (2pi, 0)

Connect the points with a smooth wave.

?

I totally get it. Thanks.

 

[firemath]

Actually, that is in radians! One half-cycle starts at 0 and ends at [MATH]\pi[/MATH], which is about 3.14.

Students often imagine that if it's in radians, it has to mention [MATH]\pi[/MATH], because that's how they always see it. But radians are just numbers. (And you can tell Desmos to measure either axis in terms of [MATH]\pi[/MATH] if you wish -- click Settings, then Step.)

Thank you for the tips.....very useful!

 

[harpazo]

Thank you for the tips.....very useful!

I second that....

 

[harpazo]

Graph two cycles of y = 3 cos x.

I know the first cycle ends at pi. Where does the second cycle end? How do I find the second cycle of the cosine function?

 

[MarkFL]

Where a cycle ends depends on where you decide it begins. A cycle is equivalent to the period, and the period of the given function is:

[MATH]T=2\pi[/MATH]
If you decide to begin at \(x=-\pi\), then the first cycle will indeed end at \(x=-\pi+2\pi=\pi\).

 

[Dr.Peterson]

In my experience, we usually think of the cycle of the cosine, like the sine, as going from 0 to 2\pi. The graph just starts at 1 and comes down, rather than starting at 0 and rising.

But as MarkFL said, you could consider a cycle to start wherever you like. If you were doing this for a teacher (or if you want to compare your answers to the book's), you have to check how they use the term.

 

[harpazo]

Where a cycle ends depends on where you decide it begins. A cycle is equivalent to the period, and the period of the given function is:

[MATH]T=2\pi[/MATH]
If you decide to begin at \(x=-\pi\), then the first cycle will indeed end at \(x=-\pi+2\pi=\pi\).

Cool. Thank you.