a)
To write: The formula for the distance between P1 and P2 .
The formula for the distance between P1 and P2 is d=|r1+r2| or d=|r1−r2| .
Given information:
The polar coordinates are (r1, θ1) and (r2, θ2) .
Formula used:
Use the law of cosines.
Calculation:
There are two cases to be considered.
- θ1−θ2 is an odd integer multiple of π
- θ1−θ2 is an even integer multiple of π
Also, use the use the Law of Cosines. It is also known that the odd integer multiple of π with cosθ is equal to −1 .
When θ1−θ2 is an odd integer multiple of π ,
d=√r21+r22−2r1r2cos(θ1−θ2)=√r21+r22−2r1r2(−1)=√r21+r22+2r1r2=√(r1+r2)2
Simplify further.
d=|r1+r2|
It is also known that the even integer multiple of π with cosθ is equal to 1 .
When θ1−θ2 is an even integer multiple of π ,
d=√r21+r22−2r1r2cos(θ1−θ2)=√r21+r22−2r1r2(1)=√r21+r22−2r1r2=√(r1−r2)2
Simplify further.
d=|r1−r2|
Therefore, the formula for the distance between P1 and P2 is d=|r1+r2| or d=|r1−r2| .
b)
To prove: That the distance between P1 and P2 is d=√r21+r22−2r1r2cos(θ1−θ2) .
Calculation:
Consider the following figure.
Find the distance d=P1P2 by using the Law of Cosines in the ΔOP1P2 .
d2=OP21+OP22−2OP1OP2cosP1OP2d2=r21+r22−2r1r2cos(θ2−θ1)d=√r21+r22−2r1r2cos(θ2−θ1)d=√r21+r22−2r1r2cos(θ1−θ2)
Therefore, it is proved that the distance between P1 and P2 is d=√r21+r22−2r1r2cos(θ1−θ2) .
c)
To find: Whether the formula in part (b) agrees with the formula(s) found in part (a)
Yes, the formula in part (b) agrees with the formulas found in part (a).
Calculation:
Consider the following formula.
d=√r21+r22−2r1r2cos(θ1−θ2)
Let θ1−θ2=kπ . Where, k is an integer.
Apply on the above formula.
d=√r21+r22−2r1r2cos(kπ)
When k=2n ,
d=√r21+r22−2r1r2cos(2nπ)=√r21+r22−2r1r2=√(r1−r2)2=|r1−r2|
When k=2n+1 ,
d=√r21+r22−2r1r2cos((2n+1)π)=√r21+r22+2r1r2=√(r1+r2)2=|r1+r2|
So the formula in part (b) agrees with the formulas found in part (a).
Therefore, yes, the formula in part (b) agrees with the formulas found in part (a).
Chapter 6 Solutions
PRECALCULUS:GRAPHICAL,...-NASTA ED.
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