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FP2 Practice Exam 1 Q3 polar co-ords

The question is:
3) The polar equation of a curve is rcos(?/2) = a for -pi<?<pi
(i) show that rsin? may be expressed as 2a(sin(?/2)) and deduce that as ? tends to pi, the curve approaches the lines with cartesian equations y=+/- 2a

so I just used double angle formulas with a bit of re-arranging, wasn't that hard, then putting pi as ? gets you to y=+/- 2a as rsin? is an expression for a y coordinate.

(ii) Sketch the curve. I have no idea how to do this. I don't even know what the expression rcos(?/2) = a is meant to represent, but I rearraged it to get r=a(sec(?/2). Then plotted r(t)=(sec(?/2) on this site:

http://www.calculator-grapher.com/graphing-calculator.html

And got pretty much a straight line through 1, although not quite.
Is this correct? Am I doing something wrong?

(iii) Calculate the area of the sector of the curve between theta = plus or minus half pi.

No idea either.... I suck at this chapter.

Thanks for any help ;>

Also how do I prove that arcsinhx = ln(x+sqrt(x^2 +1) from the starting definition of sinhx = 1/2(e^x +e^-x)
(edited 13 years ago)
Reply 1
Avatar for ztibor
ztibor
10
Original post by jamie092
The question is:
3) The polar equation of a curve is rcos(?/2) = a for -pi<?<pi
(i) show that rsin? may be expressed as 2a(sin(?/2)) and deduce that as ? tends to pi, the curve approaches the lines with cartesian equations y=+/- 2a

so I just used double angle formulas with a bit of re-arranging, wasn't that hard, then putting pi as ? gets you to y=+/- 2a as rsin? is an expression for a y coordinate.

(ii) Sketch the curve. I have no idea how to do this. I don't even know what the expression rcos(?/2) = a is meant to represent, but I rearraged it to get r=a(sec(?/2). Then plotted r(t)=(sec(?/2) on this site:

http://www.calculator-grapher.com/graphing-calculator.html

And got pretty much a straight line through 1, although not quite.
Is this correct? Am I doing something wrong?

Correct if a=1/2

(iii) Calculate the area of the sector of the curve between theta = plus or minus half pi.

No idea either.... I suck at this chapter.

Thanks for any help ;>

THE area of sector A=∫αβr2(θ) dθA=\int^{\beta}_{\alpha} r^2(\theta)\ d\theta

Also how do I prove that arcsinhx = ln(x+sqrt(x^2 +1) from the starting definition of sinhx = 1/2(e^x +e^-x)

y=1/2(e^x+e^-x)
Changing x with y
x=1/2(e^y+e^-y) and solve this to y (=arsinhx)
Reply 2
Avatar for jamie092
jamie092
OP
11
thanks so much <3

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