Question

Freddie is at chess practice waiting on his opponent's next move. He notices that the 4-inch-long minute hand is rotating around the clock and marking off time like degrees on a unit circle.

Part 1: How many radians does the minute hand move from 3:35 to 3:55? (Hint: Find the number of degrees per minute first.)
Part 2: How far does the tip of the minute hand travel during that time?
Part 3: How many radians on the unit circle would the minute hand travel from 0° if it were to move 3π inches?
Part 4: What is the coordinate point associated with this radian measure?

Answer 1

I found 2π/3 for part 1

Answer 2

8π/3 for part 2

Answer 3

I also found 3π/4 for part 3, I just need help with part 4

Answer 4

@SolomonZelman

Answer 5

@beccaboo333

Answer 6

I'm not very good with math ;-;

Answer 7

Ok sorry

Answer 8

@mathmale

Answer 9

Would you explain in your own words what you are being asked for in Part 4?
What mathematical terms apply?

Answer 10

I am supposed to use the radian measure that I found in part 3 to find the coordinates of where the minute hand is supposed to be at that measure, but I'm not sure how.

Answer 11

Good. We need to use the circumference formula. You familiar with that?

Answer 12

Yes I am. C=πd

Answer 13

Good. What about the arc length formula? Type that out, please.

Answer 14

Excuse me, but if you've done Part 3 already, you probably already have some of the answer for Part 4. What is your answer for part 3, and how did y ou obtain it?

Answer 15

Part 3: How many radians on the unit circle would the minute hand travel from 0° if it were to move 3π inches?

Answer 16

Arc length is s=r(theta)
I worked backwards from the arc length formula

Answer 17

I did 3π=4(theta)

Answer 18

And solved for theta

Answer 19

The tip of the minute had moves 3pi radians along the circle of radius 4 inches. Fair enough: I agree with your logic.

Answer 20

What is your value for this angle, theta? Are you measuring it in degrees or in radians?

Answer 21

I got 3π/4 radians, which is 135 degrees.

Answer 22

It asks for radians in part 3.

Answer 23

Looks great.
So now, the minute hand has rotated (3/4)pi radians. The length of the minute hand is 4 inches.
Do these formulas look at all familiar to you?

x=r cos theta

y=r sin theta

Answer 24

As a footnote, the cosine function is defined as the length of the side adjacent to the angle in a right triangle, divided by the hypotenuse of the triangle. Familiar?

Answer 25

No, I'm sorry

Answer 26

The second part is familiar.

Answer 27

The footnote, I mean

Answer 28

Have you studied trigonometry yet?
Trig presents the easiest method of coming up with the answer to Part IV.

Answer 29

Yes I have

Answer 30

then you know that \[\sin \theta=\frac{ opp }{ hyp }\] where "opp" represents the length of side opposite the angle in question, and "hyp" is (obviously) the hypotenuse length.

Answer 31

Yes I do

Answer 32

If\[\sin \theta=\frac{ opp }{ hyp }\]then opp=(hyp)(sin theta).

Answer 33

\[y=r*\sin \]

Answer 34

sorry. I mean: y = r * sin theta.

Answer 35

How do you define the cosine function?

Answer 36

Do you mean cos=adj/hyp or f(x)=a cos(bx-c)+d ?

Answer 37

The first formula, that's the definition of the cosine function.

If y ou accept that definition, and if you multiply both sides by "hyp," you obtain x= (hyp)(cos theta).

Here, your radius is 4 inches, your angle is (3/4)pi or 135 degrees.

First find the x-coordinate of the tip of the minute hand by finding x=(4 inches)(cosine 135 deg).

Answer 38

Is x the x-coordinate that I'm looking for?

Answer 39

Yes. You want the coordinates of the tip of the minute hand, and are now finding the x-coord. separately. Next, you'll find the y-coord.

Answer 40

Would the x-coordinate be -2.8?

Answer 41

Yes! Now, try finding the y-coordinate. Would you expect it to be positive or negative? Why? Hint: in which direction does the minute hand turn?

Answer 42

It goes clockwise so the y-coordinate would be negative.

Answer 43

You're doing very well indeed. Indeed the tip of the minute hand would be in quadrant III as measured counterclockwise from 0 degrees or 0 radians.

What do you now think are the coordinates of the tip of the minute hand?

Answer 44

I'm sorry, sir. I'm not sure about what to do from here.

Answer 45

You've correctly found the x-coordinate of the tip of the minute hand: x=-2.8.
You've correctly predicted that the y-coord. is negative.
By evaluating (4 inches)(sin [-(3/4)pi], you'll end up with the y-coordinate. What is it?

Answer 46

I got -0.16 but I don't think that I'm right.

Answer 47

How did you obtain that? Are you using a calculator?

Answer 48

Yes I am

Answer 49

I used sin for 3/4, then multiplied it by pi, and then multiplied it by 4.

Answer 50

Can you set it to either degree or radian mode?
if so, set it to degree mode if you're finding y=sin (-135 deg) or to radian mode if you're finding y = sin ([3/4]pi).

Answer 51

I see. That explains what happened here. FIRST, multiply (-3/4) by pi. only AFTER that, find the sine of the result.

Answer 52

Please try again.

Answer 53

In which mode (degrees or radians) is your calculator currently operating?

Answer 54

Degree

Answer 55

then the angle in question is -135 degrees. type in cos (-135) and see what you get.

Answer 56

I got -0.707

Answer 57

And that's correct.

Answer 58

For the y-coordinate?

Answer 59

Your -0.707 is correct as the value of the sine of (-135 deg). But you still have to multiply that by 4 inches, don't you? Try it, please.
Hint: the x- and the y- coordinates happen to be the same here!

Answer 60

Oh wow, I kept getting -2.8 for y but I thought I was wrong

Answer 61

So the coordinates are (-2.8.-2.8)?

Answer 62

Yes, that's perfect, EXCEPT you'll need to write "inches" after each of the two coordinates.

Answer 63

Ok. Thank you, sir, so very much!

Answer 64

My great pleasure, fawkes! Talk with you again!