Edexcel A-level Further Maths coverage

Polar coordinates

Section CP-7
3 spec leafs

Notes and three levels of exam-style practice for each registered specification leaf in this section.

Open the printable pack
CP-7.1

Understand and use polar coordinates and be able to convert between polar and Cartesian coordinates.

  • Polar coordinates (r,θ)(r,\theta) locate a point a directed distance rr from the pole at angle θ\theta measured anticlockwise from the initial line.
  • Convert to Cartesian coordinates using x=rcos⁡θx=r\cos\theta and y=rsin⁡θy=r\sin\theta; conversely, r2=x2+y2r^2=x^2+y^2 and tan⁡θ=y/x\tan\theta=y/x with the quadrant checked.
  • To convert a polar curve, multiply by rr when useful and replace rcos⁡θr\cos\theta by xx, rsin⁡θr\sin\theta by yy and r2r^2 by x2+y2x^2+y^2.
  • Polar coordinates are not unique: (r,θ)(r,\theta) and (−r,θ+π)(-r,\theta+\pi) represent the same point. A common error is to use arctan⁡(y/x)\arctan(y/x) without correcting its quadrant.

Tier 1 · Easy

2 marks
ORIGINAL

Convert the polar coordinates (4,π/6)(4,\pi/6) to Cartesian coordinates.

Tier 2 · Standard

3 marks
ORIGINAL

Express the Cartesian point (−3,33)(-3,3\sqrt3) in polar form, taking r>0r>0 and 0≤θ<2π0\leq\theta<2\pi.

Tier 3 · Hard

5 marks
ORIGINAL

Convert the polar curve r=4cos⁡θ+2sin⁡θr=4\cos\theta+2\sin\theta to Cartesian form and identify the curve.

CP-7.2

Sketch curves with r given as a function of theta, including use of trigonometric functions.

  • Build a polar sketch from symmetry, zeros of rr, maxima and minima of rr, and values on the initial line; plot negative rr in the opposite direction.
  • If replacing θ\theta by −θ-\theta leaves the equation unchanged, the curve is symmetric about the initial line; related tests identify symmetry about θ=π/2\theta=\pi/2 or the pole.
  • Curves such as r=acos⁡θr=a\cos\theta and r=asin⁡θr=a\sin\theta are circles, while r=asin⁡(nθ)r=a\sin(n\theta) and r=acos⁡(nθ)r=a\cos(n\theta) generate rose curves whose petals follow the extreme values of rr.
  • Tangents parallel to the initial line occur where ddθ(rsin⁡θ)=0\frac{d}{d\theta}(r\sin\theta)=0, and tangents perpendicular to it where ddθ(rcos⁡θ)=0\frac{d}{d\theta}(r\cos\theta)=0. A common error is to discard negative values of rr, which can remove an entire loop or petal.

Tier 1 · Easy

2 marks
ORIGINAL

Sketch the polar curve r=3r=3, labelling its key geometric features.

Tier 2 · Standard

4 marks
ORIGINAL

Sketch the polar curve r=2cos⁡θr=2\cos\theta. State its Cartesian equation, centre and radius.

Tier 3 · Hard

6 marks
ORIGINAL

For 0≤θ<2π0\leq\theta<2\pi, sketch the complete polar curve r=4sin⁡(2θ)r=4\sin(2\theta). Label the directions and lengths of all petals and the values of θ\theta at which the curve passes through the pole.

CP-7.3

Find the area enclosed by a polar curve.

  • The area swept by a polar curve from θ=α\theta=\alpha to θ=β\theta=\beta is A=12∫αβr2 dθA=\dfrac12\int_\alpha^\beta r^2\,d\theta.
  • Find the angular limits from intersections, zeros of rr or symmetry before integrating; a complete loop may occupy only part of a full turn.
  • For the area between two polar curves over the same angles, integrate 12(router2−rinner2)\frac12(r_{\text{outer}}^2-r_{\text{inner}}^2).
  • Sketch and shade the intended region first. Common errors are omitting the factor 1/21/2, using rr instead of r2r^2, or counting a symmetric loop twice.

Tier 1 · Easy

3 marks
ORIGINAL

Find the exact area of the sector enclosed by r=2r=2 and the rays θ=0\theta=0 and θ=π/3\theta=\pi/3.

Tier 2 · Standard

5 marks
ORIGINAL

Determine the exact area enclosed by the loop r=3sin⁡θr=3\sin\theta.

Tier 3 · Hard

7 marks
ORIGINAL

The curves r=2r=2 and r=4cos⁡θr=4\cos\theta enclose a region that lies inside r=4cos⁡θr=4\cos\theta but outside r=2r=2. Find its exact area.