Sinusoidal Model(I think)

[vale saffron]

I’m not sure where to start with this I’m fairly certain I’m 100% wrong

[tribal wedge]

How can you be wrong if you haven't started?

[vale saffron]

I did start

I just don’t trust my answers

[tribal wedge]

,rotate

[vale saffron]

Apologies for my terrible handwriting

[tribal wedge]

What's d(0)?

[vale saffron]

I need to figure out the value of d

[tribal wedge]

...d is a function...

[vale saffron]

Oh right

I keep thinking of p q and r as a b and d

[tribal wedge]

So what's d(0)?

[vale saffron]

Wouldn’t that be 0

Since nothing has happened at that point

And yes I’m fully aware I probably sound stupid right now

[tribal wedge]

Not necessarily. Read the question thoroughly. What is d(t) measuring?

[vale saffron]

How far the cyclist has moved in a given amount of time in seconds

[tribal wedge]

Read thoroughly.

[vale saffron]

Ok

It’s how many meters north of the east west diameter of the circuit in seconds the cyclist is

[tribal wedge]

And where do they start?

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Due east

Of the center

[tribal wedge]

Which is how many meters north of the east/west diameter?

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Would it be 40

I’m fairly sure it’s not

I speak before I think

[tribal wedge]

You might want to... not do that.

[vale saffron]

I’m a little impulsive for my own good , my bad

I’m just trying to do well in this class but I don’t get some of the content

[tribal wedge]

Okay, you know what? Draw a picture. Diagrams are always useful when possible.

[vale saffron]

A circle diagram I assume

That’s what I’m thinking anyways

[tribal wedge]

The problem is about a circle, so...

[vale saffron]

Ok

Got it

Next step is… I don’t know

[vale saffron]

^

[tribal wedge]

Show me what you've drawn so far.

[vale saffron]

Literally just a circle

[tribal wedge]

So now just put all the information from the problem into the diagram.

Draw the east/west diameter, since that's the one you care about, and note that it's 80 meters. Then draw a point where the cyclist starts.

[vale saffron]

;rotate

,rotate

[vale saffron]

Like this?

[tribal wedge]

Is that the east/west diameter? Because it looks like the north/south diameter.

And why is "80m" on the circumference of the circle?

[vale saffron]

,Roste

,rotate

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Where does the cyclist start?

[vale saffron]

Around where the arrow is

I think

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Draw the point.

[vale saffron]

I’m probably wrong

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I... don't know why the point isn't on the circumference of the circle...

[vale saffron]

^

That better?

[tribal wedge]

Sure. So d(0) is what?

[vale saffron]

I’m not sure

[tribal wedge]

How many meters is the point you just drew above the line you just drew?

[vale saffron]

0

I believe

[tribal wedge]

So d(0) = 0.

[vale saffron]

Yes

[tribal wedge]

d(0) = p * sin(0 * q) + r
= p * sin(0) + r
= p * 0 + r
= r```
So `r = 0`.

[vale saffron]

Ok

So now I have to find p

I think I got q already

[tribal wedge]

I doubt it.

[vale saffron]

Correct me if I’m wrong but, the period is 20 seconds, 20=360/b
Multiply by b on both sides
20b=360
Divide by 20
B=18

[tribal wedge]

Whoa, 360? We're working in degrees?

[vale saffron]

That’s how finding the q is for me

It’s stuff like
d+a is max
d-a is min
a is amplitude

Stuff like that

But the symbols got changed a little bit is all

Here’s the base function
h(t)=asin(bt)+d

It can be the same but instead of sin it’s cos

[tribal wedge]

Here's how I do it: d(t) = d(t + 20) p sin(qt) + r = p sin(q(t + 20)) + r sin(qt) = sin(qt + 20q) qt + 2npi = qt + 20q 20q = 2npi q = npi/10
n is an integer.

[vale saffron]

Oh god

[tribal wedge]

And I'm pretty sure n = 1.

[vale saffron]

n isn’t a variable in this though,

?

[tribal wedge]

Look, you got the right answer. pi/10 rad = 18 deg.

[vale saffron]

It’s just p q r

Cool

Now I just need to find p

And then solve part b

I wish I had even half the math smarts you have

[tribal wedge]

Radians have a periodicity of 2pi, so any measure of radians can be shifted by an integer multiple of 2npi without affecting the actual angle measured.

It's like saying an angle of 18 degrees is the same as an angle of 378 degrees.

[vale saffron]

I’m not sure what you’re getting at

To be completely honest

[tribal wedge]

I'm not surprised.

Like, imagine you have an angle of 18 degrees, right?

[vale saffron]

Ok

[tribal wedge]

Now imagine you rotated one arm of that angle 360 degrees. Does that change the actual angle between the arms?

[vale saffron]

No

Because it did a full rotation and ended in the same spot, right?

[tribal wedge]

So an angle of 18 degrees is the same as an angle of 18 + 360 = 378 degrees.

[vale saffron]

Ok

Took me a second

[tribal wedge]

And in radians, a full rotation is 2pi.

[vale saffron]

Ok

[tribal wedge]

Anyway.

So finding p is actually pretty simple.

[vale saffron]

Alright

[tribal wedge]

What is d(5)?

I just realized something.

We're never told if the cyclist is riding clockwise or counterclockwise.

[vale saffron]

d(5)=psin(18t)+0

[tribal wedge]

I also realized you drew the cyclist due west, not due east, of the center.

[tribal wedge]

No.

In... more than one way.

[vale saffron]

In more than one way I’m wrong?

[tribal wedge]

For one, you forgot to replace t with 5.

[vale saffron]

It’d be 90 then

d(5)=psin(90)+0

[tribal wedge]

For two, when I ask you what d(a) is, I'm not asking you to evaluate d at a. I'm asking you what it should be based on the parameters of the question. What do we want d(5) to be?

[vale saffron]

I’m not sure

[tribal wedge]

Why did I say 5?

[vale saffron]

Because I want to find out how far the cyclist has gone in 5 seconds

[tribal wedge]

But why 5 seconds? Why not 6 or 4 or 17?

[vale saffron]

Because that’s the value that will give the answer to the value of p?

[tribal wedge]

But why?

Maybe now is a good time to look at what d(5) actually is in our function.

[vale saffron]

Ok then

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Look at this. What's special about this?

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It’s a function for figuring out the value of how many meters have been traveled in 5 seconds?

[tribal wedge]

...what's sin(90 deg)?

[vale saffron]

1

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That's what's special.

[vale saffron]

I don’t understand the “special” thing

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Is sin(x) ever greater than 1?

[vale saffron]

I don’t think so

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Then at 5 seconds, the cyclist is as far from the diameter as they can possibly be.

[vale saffron]

Because it has to have another value after it to increase its value

[tribal wedge]

And how far is that?

[vale saffron]

20 meters?

[tribal wedge]

Show your work on that one.

[vale saffron]

Probably not

I would say you’ll think im stupid but im pretty sure I’ve already given that impression

[tribal wedge]

I don't necessarily think you're stupid, I think you just don't think enough.

A habit I hope to do my part in breaking.

[vale saffron]

I think it’s more based on the fact I haven’t seen a question in this topic structures like this before

*structured

[tribal wedge]

Okay, but you've clearly been working on this topic for a bit. This is like a mid-unit question.

[vale saffron]

I’ve been learning this topic for two days

[tribal wedge]

...okay, like in school or on your own?

[vale saffron]

School

Most of the time it made sense it’s just the way this question is structured makes it hard to work around for me

[tribal wedge]

Anyway, I don't really see how not being used to trigonometry word problems affects your geometry abilities.

[vale saffron]

It’s been a while since I last learned geometry, but enough about my excuses , any hints that may help me with this case

[tribal wedge]

What's the diameter of the circle?

[vale saffron]

80 meters

That’s pretty much all I know

[tribal wedge]

Are you kidding? That's everything.

[vale saffron]

Everything I need to know?

[tribal wedge]

A circle is defined - literally defined - by its radius. As in, the literal definition of a circle is "all points that are a distance r from a center point". So if you know the radius of a circle, you know literally everything about it.

[vale saffron]

I forget if it’s radius or diameter is the larger number

Oh wait wasn’t it that diameter is radius squared

[tribal wedge]

...what's the definition of the radius?

[vale saffron]

So I need to find the square root of 80

[tribal wedge]

What is the definition of the radius?

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From the middle to the outer edge of a given circle

[tribal wedge]

...the length of the line from the center to the circumference. And what is the definition of the diameter?

[vale saffron]

Straight line passing from side to side

Through the center of a circle

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So logically, the diameter would be what in terms of the radius?

[vale saffron]

Two times the radius

[tribal wedge]

Right.

[vale saffron]

And the radius would be 40

[tribal wedge]

You could also have got there by dimensional analysis.

[vale saffron]

I don’t think like that sadly

[tribal wedge]

Radius and diameter are both lengths, but radius squared is an area.

[vale saffron]

Ok

But I’m not looking to square the radius I believe

[tribal wedge]

Yes, and I was trying to explain to you how you can figure that out for yourself next time.

[vale saffron]

Sorry

[tribal wedge]

Anyway. This is correct. So what is the maximum possible value of d(t)?

[vale saffron]

20

Because it takes 20 seconds per lap of the full track

[tribal wedge]

...what is d(t) measuring again???

[vale saffron]

How many meters north of the east west diameter of the circuit

[tribal wedge]

So what is the maximum that can be?

[vale saffron]

1

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Why?

[vale saffron]

… wait I think that’s wrong

Sin(90)

[tribal wedge]

I need you.

To think logically here.

Don't just make random guesses.

[vale saffron]

Ok

[tribal wedge]

Have reasons for the things you say and tell me them in explicit detail.

[vale saffron]

Ok

[tribal wedge]

Is the cyclist ever not on the circle?

[vale saffron]

The cyclist is always on the circle

I got that wrong didn’t I

[tribal wedge]

So logically, how far away is it possible for the cyclist to be from the diameter of the circle?

[vale saffron]

Well if they’re always on the circle, and unable to leave it,, the farthest they can be from the diameter is 40 because that is the farthest point from the center without leaving the circle

Because of the radius

[tribal wedge]

Very good.

So that's p, but now we have a choice.

[vale saffron]

So p is 40

[tribal wedge]

Almost.

Like I said, we have a choice. Or rather, a piece of information the question didn't give us.

[vale saffron]

40 is just part of solving for p

Ok

[tribal wedge]

The fact is, p = 40 if the cyclist is riding counterclockwise. If they're riding clockwise, however, then p = -40.

And we aren't told which direction they're riding in.

[vale saffron]

Ok

So I guess at this point I try plugging in both into the equation to see which answer fits best with aforementioned data?

[tribal wedge]

No.

What did I just say? They both fit the question because the piece of information that would tell us which one to use isn't in the question.

[vale saffron]

alright

[tribal wedge]

So just pick one, and then note which direction of travel you're assuming.

[vale saffron]

I just think it’d be clockwise

[tribal wedge]

P = -40 it is.

[vale saffron]

So it’s be -40

[tribal wedge]

Make sure to note that you're assuming clockwise travel because you weren't told which direction in the question.

[vale saffron]

Ok

I think I denoted that with my diagram arrow

[tribal wedge]

I think you should state it explicitly. If you don't tell your teacher that they didn't give you the information, they might think they did and mark you wrong.

[vale saffron]

I don’t know if this is gonna make you mad or not, but the teacher grades more based on completion

But my teacher last year made me into a wannabe perfectionist so I kinda strive to get everything right

[tribal wedge]

That actually makes a lot of sense. I thought this question was a bit too advanced for this early in the unit, but it's one you're not expected to get right, it's one you're expected to try so that it can be explained to you in class tomorrow.

[vale saffron]

That’s assuming I even go tomorrow, I’m kinda sick right now :/

[tribal wedge]

Then it's a good thing I'm here to explain it to you tonight. So, in the end, what's d(t)?

[vale saffron]

d(t)=-40sin(18t)+0

[tribal wedge]

You can drop the +0.

[vale saffron]

I just do it as a memory aide

[tribal wedge]

It's + 0

[vale saffron]

Helps me memorize the equation itself

[tribal wedge]

Alrighty, then... on to b?

[vale saffron]

Yep

[tribal wedge]

So what exactly is b asking?

[vale saffron]

How long it takes for the cyclist to be 10 meters north of the east west diameter

*south not north

[vale saffron]

^

[tribal wedge]

Or, in other words?

[vale saffron]

I’m not sure

[tribal wedge]

We want the smallest t for which d(t) = -10.

[vale saffron]

Ok

[tribal wedge]

So! d(t) = -10 -40sin(18t deg) = -10 sin(18t deg) = 1/4

[vale saffron]

Kinda hard to focus rn my dog is pestering me

Would you kindly explain to me in simple terms what is being asked of me at this point in time?

[tribal wedge]

Sine of what equals 1/4?

[vale saffron]

Roughly sin(15)

[vale saffron]

^

That looks about right,

[tribal wedge]

Roughly?

[vale saffron]

?

It’s the best answer I could come up with

[tribal wedge]

Just use the inverse sine function on your calculator.

[vale saffron]

Where would I find that

I’m using a ti-84 CE

[tribal wedge]

I need a picture.

[vale saffron]

Sorry if it’s shaky my dog is on top of me and shaking because of some thunder outside

[vale saffron]

^

[tribal wedge]

See above the sin key where there's sin^-1 in blue?

[vale saffron]

Yes

[tribal wedge]

That means you hit the blue key, then the sine key.

[vale saffron]

Ye

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^

[tribal wedge]

...what?

[vale saffron]

What was I supposed to put in for the value

1/4?

[tribal wedge]

...duh?

[vale saffron]

I got 14.477

[tribal wedge]

Okay, so that's equal to 18t.

[vale saffron]

And do I need anything after that?

Or is that all

[tribal wedge]

... Solve the equation.

[vale saffron]

t=0.804

[tribal wedge]

There you go.

[vale saffron]

Cool