Question

Show that the distance (d) between two points......

Answer 1

Show that the distance between two points \[P _{1} (r _{1}, \theta _{1})\] and
\[P _{2}(r _{2},\theta _{2})\] is given by:
\[d=\sqrt{r _{1}^2 + r _{2}^2-2r_1r_2\cos(\theta_1-\theta_2)}\]

Answer 2

@wio

Answer 3

The algebraic solution is to use \(x=r\cos\theta\) and \(y=r\sin\theta\).

Answer 4

so do you have to plug into each x and y?

Answer 5

im kind of confused

Answer 6

Yeah, and then use the distance formula for Cartesian coordinates.

Answer 7

\[
d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}
\]

Answer 8

do you plug in rcostheta for both x1 and x2?

Answer 9

Well, you want to use subscripts

Answer 10

\[
P_1(r_1,\theta_1) \to (x_1,y_1)
\]

Answer 11

wait so for x1 do you plug in r1 or r1costheta

Answer 12

\[
x_1 = r_1\cos(\theta_1)
\]

Answer 13

and x2 just replace all the subscripts with 2?

Answer 14

soo...

Answer 15

yeah

Answer 16

and is y=rsintheta

Answer 17

with both subscripts?

Answer 18

yes

Answer 19

im stuck at \[d=\sqrt{(r_2\cos \theta _2-r_1 \cos \theta _1)^2+(r_2\sin \theta_2-r_1\sin \theta _1)^2}\]

Answer 20

You need to expand, and remember your trig identities

Answer 21

can you do the parenthesis with the cosine one so i get how to do the other parenthesis?

Answer 22

Do you know how to foil? You have to foil first.

Answer 23

i know how to foil its just i get a little confused sometimes: for example,
\[(-r_1\cos \theta _1) * (r_2\cos \theta _2)\] how would i do this?

Answer 24

is it
\[-r_1r_2\cos^2 \theta_1 \theta _2\]

Answer 25

Remember that the answer has: \[
\cos(\theta_1-\theta_2)
\]Do you remember the trig identify for sum of angles and cosine?

Answer 26

i think so

Answer 27

actually no, ithought i did

Answer 28

but we did learn it its just i forgot

Answer 29

\[
\cos(\theta_1-\theta_2) = \cos(\theta_1)\cos(\theta_2)- \sin(\theta_1)\sin(\theta_2)
\]

Answer 30

but i dont have the sin theta yet

Answer 31

look for it

Answer 32

im talking about those two things i was multiplying

Answer 33

how would you multiply those two first

Answer 34

\[(-r_1\cos \theta _1) \cdot (r_2\cos \theta _2) = -r_1r_2\cos(\theta_1)\cos(\theta_2)\]

Answer 35

it doesn't simplify any more than that.

Answer 36

ok

Answer 37

now im doing the sin side