Question

SUBJECT TO EDITING.
Trig questions:

#1: Use 3.1416 for π unless your calculator has a key marked π.
Use a calculator to convert 1' (1 minute) to radians to three significant digits.

#2: Write the angle as a difference involving 2π. For example, 5π/3 = 2π − π/3.
7π/4

#3: If a central angle with its vertex at the center of the earth has a measure of 1', then the arc on the surface of the earth that is cut off by this angle (known as the great circle distance) has a measure of 1 nautical mile (see the figure below).
Find the number of regular (statute) miles in 1 nautical mile to the nearest hundredth of a mile. (Use 4,000 miles for the radius of the earth.)

Answer 1

\[1'={\frac{1}{60}}^{\circ}\]
\[180^{\circ}=\pi \space rad\]\[1^{\circ}=\frac{\pi}{180} \space rad\]\[{\frac{1}{60}}^{\circ}=?? \space rad\]

Answer 2

What do you think?

Answer 3

I get 0.0002908...

Answer 4

How much you are getting?

Answer 5

Let me check

Answer 6

Okay.

Answer 7

and what is the answer they've given?

Answer 8

Not mentioned.

Answer 9

I mean, it says "3 sig figs" and that would give me basically 0?

Answer 10

Your answer is absolutely correct, you can eve google "minutes to radian conversion"
I think zeros after a decimal don't count as significant unless they are after some other number, I might be remembering that wrong though

Answer 11

So... scientific notation?

Answer 12

What exactly did you write for your answer?

Answer 13

I'd done it wrong before lol

Answer 14

So your found your mistake?

Answer 15

No... I'm not sure if this is the right answer, so I'd have to check and make sure.
I'm not sure how many attempts I have left x_x

Answer 16

Oh, if you are giving an online test/exam, they sometimes require you to write in a particular way, but mathematically your answer is 100% correct

Answer 17

*puts in 0.000291*
*gets right answer*
WHAT.

Answer 18

THE HECK.

Answer 19

I see your mistake now, why are you writing 2.91??
Write the answer that we found!
0.000290 - upto 3 significant figures!

Answer 20

291. I had to round.

Answer 21

It's right.

Answer 22

ahh, see

Answer 23

I'd put some entirely diff # before. lol

Answer 24

anyways for your next part we have

Answer 25

2π+3π/4 didn't work

Answer 26

You want to express the following fraction in terms of 2pi
\[\frac{\pi}{10800}\]
So if you were to somehow split the numerator to express it as a sum of 2 fractions, you'd want 1 of the fractions to have a numerator of 2 times the denominator so that you get 2pi upon cancelling
\[\frac{\pi}{a}=\frac{\pi(1)}{a}=\frac{\pi(1+b-b)}{a}=\frac{b \pi+(1-b)\pi}{a}=\frac{b \pi}{a}+\frac{(1-b)\pi}{a}\]
b is any arbitrary constant, we will choose b such that
\[b=2a\]
So substituting we get
\[\frac{2a \pi}{a}+\frac{(1-2a)\pi}{a}=2\pi+\frac{(1-2a)\pi}{a}\]
Here we have
\[a=10800\]

Answer 27

Did you get all that?

Answer 28

Oh dear

Answer 29

10800??

Answer 30

yep
Here we have
a=10800
\[{\frac{1}{60}}^{\circ}=\frac{\pi}{180} \times \frac{1}{60} \space rad\]

Answer 31

why is a 10800?

Answer 32

ok first do you understand that
\[1^{\circ}=\frac{\pi}{180} \space rad\]

Answer 33

Yes.

Answer 34

So now we want to find
\[1'={\frac{1}{60}}^{\circ}\]
How would you do that?you'd divide by 60 of course!
Don't worry, I said that oh dear because you were lagging and I was afraid you may have to leave the question in between

Answer 35

oh lol.

Answer 36

And yes.

Answer 37

Now read my explanation above of how to convert it into terms of 2pi, i've written it above

Answer 38

I got lost.

Answer 39

o-o

Answer 40

Ok i'll try again

Answer 41

Suppose you have a fraction of the form
\[\frac{\pi}{a}\]

Answer 42

You can multiply it with 1, makes no difference, I'll write that just to create some clarity
\[\frac{\pi(1)}{a}\]

Answer 43

following so far?

Answer 44

Got it so far.

Answer 45

Now, we can add and subtract any number, it will make no difference
Suppose we have some equation
\[x^2+2x+3=9\]
If we add and subtract say, root 7, it will make no difference to the equation overall
\[x^2+2x+3+\sqrt{7}-\sqrt{7}=9\]
Similarly we add and subtract an arbitrary constant b
\[\frac{\pi(1+b-b)}{a}\]
We can take b as whatever we want, but there's a particular value of b we desire, I'll show you what value later

Answer 46

so far good?

Answer 47

Yep

Answer 48

To further illustrate, I'll distribute the pi and see what we get
\[\frac{\pi(1+b-b)}{a}=\frac{\pi+b \pi-b \pi}{a}=\frac{\pi}{a}\]

Answer 49

So it makes no difference

Answer 50

Sorry; I'm tired so I take a while to process this stuff haha

Answer 51

Next we will split our fraction into 2 fractions
\[\frac{\pi(b+1-b)}{a}=\frac{\pi b+\pi(1-b)}{a}=\frac{\pi b}{a}+\frac{\pi(1-b)}{a}\]

Answer 52

Just keep responding if you are following, when you're not, let me know

Answer 53

Okay.

Answer 54

Actually... um, can you just tell me what's wrong with the equation I put in?

Answer 55

Now b can take value we want, right? This is because we can add AND subtract any number from an equation or an expression at the same time
So if we were to let
\[b=2a\]
That is completely valid and allowed, like I said b can take any value

Answer 56

Sure, I'll see

Answer 57

Where's your attempt ??

Answer 58

Eh ?

Answer 59

You want me to check what you've done wrong for question 2, right?so let me see your work

Answer 60

http://icecream.me/3db4b88a7128b83b79b4db0bb933e6c5

Answer 61

Oh I see, so they want you to express
\[\frac{7\pi}{4}\]
I thought they meant the answer you got from 1st part

Answer 62

You've expressed it in terms of pi not 2pi

Answer 63

What number do you think when divided by 4 gives 2?

Answer 64

?

Answer 65

\[\pi=\frac{4\pi}{4}?\rightarrow \pi+\frac{3\pi}{4}=\frac{4\pi}{4}+\frac{3\pi}{4}=\frac{7\pi}{4}\]

Answer 66

You want to express it in terms of 2pi
you are expressing in terms of pi
Answer me and you'll solve it, what number when divided by 4 will give u 2?

Answer 67

I want to express it in terms of pi?

Answer 68

Nope, check your question

Answer 69

!
FFFFFF

Answer 70

So then:\[2\pi-\frac{\pi}{4}\]

Answer 71

yep!

Answer 72

BAH I feel dumb

Answer 73

What about part 3?

Answer 74

You didn't post any part 3

Answer 75

oh I see it now

Answer 76

You sure? :p

Answer 77

xD

Answer 78

ok so first we have
\[r=4000 \space mile\]\[l=1 \space nautical \space mile\]\[\theta=1'=\frac{\pi}{10800} \space rad\]
We have the formula

\[l=r \theta\]
Where, theta is in radians, very important!
so we get
\[1 \space nautical \space mile = 4000 \times \frac{\pi}{10800} \space miles\]
We already calculated the value of the angle earlier in radians it was about
\[\theta \approx 0.000291\]\[1 \space nautical \space mile = 4000 \times 0.000291 \space miles\]