Further Mathematics · AQA 7367 · Topic G

Polar coordinates

Describing points by a distance r and an angle θ instead of x and y. You sketch curves like r = 1 + cos θ, convert to and from Cartesian form, and find areas inside curves and tangents.

  • 3 sub-topics
  • 2 in Year 12, 1 in Year 13
  • 49 practice questions
  • 3 animated lessons
Practice questions (49) Lessons Worksheets All AQA Further Mathematics topics

What the specification says

Every sub-topic AQA can examine in this topic, with its reference number from the official specification. The Y12 and Y13 tags show when each sub-topic is usually taught.

  1. G1Converting between polar and Cartesian coordinatesY12
  2. G2Sketch curves with r given as a function of θY12
  3. G3Areas enclosed by polar curvesY13

Try 3 questions

Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 49 questions in this topic, free, no account needed.

Polar coordinates

Not needed for Year 12Easy

Polar to Cartesian · Year 13 · 2 marks

  1. Convert the polar point (4,π3)\left(4, \dfrac{\pi}{3}\right) to Cartesian coordinates.
    (2)
Show worked solutionHide worked solution
  1. Use x=rcos⁡θx = r\cos\theta and y=rsin⁡θy = r\sin\theta.xx=4cos⁡π3{}= 4\cos\dfrac{\pi}{3}=2, y{}= 2,\ y=4sin⁡π3{}= 4\sin\dfrac{\pi}{3}=23{}= 2\sqrt{3}

Total2

  • M
  • A
  • B
How did it go?

Report a problem with this question

Polar coordinates

Not needed for Year 12Medium

Area inside a polar curve · Year 13 · 5 marks

A curve has polar equation r=2(1+cos⁡θ)r = 2(1 + \cos\theta), 0≤θ<2π0 \le \theta < 2\pi.
  1. Find the exact area enclosed by the curve.
    (5)
Show worked solutionHide worked solution
  1. Area in polar coordinates is 12∫r2 dθ\tfrac{1}{2}\int r^2\,\mathrm{d}\theta over one full turn.AA=12∫02π4(1+cos⁡θ)2 dθ{}= \tfrac{1}{2}\int_0^{2\pi} 4(1 + \cos\theta)^2\,\mathrm{d}\theta
  2. Expand the bracket before integrating.=2∫02π(1+2cos⁡θ+cos2⁡θ)dθ= 2\int_0^{2\pi} \left(1 + 2\cos\theta + \cos^2\theta\right)\mathrm{d}\theta
  3. cos2⁡θ\cos^2\theta cannot be integrated directly, so use the double angle formula.cos2⁡θ\cos^2\theta=12(1+cos⁡2θ){}= \tfrac{1}{2}(1 + \cos 2\theta)
  4. Integrate each term.=2[32θ+2sin⁡θ+14sin⁡2θ]02π= 2\left[\tfrac{3}{2}\theta + 2\sin\theta + \tfrac{1}{4}\sin 2\theta\right]_0^{2\pi}
  5. The sine terms are zero at both limits.=6π= 6\pi

Total5

  • M
  • A
  • B
How did it go?

Report a problem with this question

Polar coordinates

Not needed for Year 12Hard

Tangents parallel to the initial line · Year 13 · 5 marks

  1. Find the points on r=1+cos⁡θr = 1 + \cos\theta, 0<θ<π0 < \theta < \pi, where the tangent is parallel to the initial line.
    (5)
Show worked solutionHide worked solution
  1. A tangent parallel to the initial line is where yy is stationary.yy=rsin⁡θ{}= r\sin\theta=sin⁡θ{}= \sin\theta+sin⁡θcos⁡θ{}+ \sin\theta\cos\theta
  2. sin⁡θcos⁡θ\sin\theta\cos\theta=12sin⁡2θ{}= \tfrac{1}{2}\sin 2\theta.dydθ\dfrac{\mathrm{d}y}{\mathrm{d}\theta}=cos⁡θ+cos⁡2θ{}= \cos\theta + \cos 2\theta
  3. Set to zero and use cos⁡2θ=2cos2⁡θ−1\cos 2\theta = 2\cos^2\theta - 1.2cos2⁡θ+cos⁡θ−12\cos^2\theta + \cos\theta - 1=0{}= 0⇒(2cos⁡θ−1)(cos⁡θ+1){}\Rightarrow (2\cos\theta - 1)(\cos\theta + 1)=0{}= 0
  4. cos⁡θ=−1\cos\theta = -1 gives θ=π\theta = \pi, outside the range.cos⁡θ\cos\theta=12{}= \tfrac{1}{2}⇒θ{}\Rightarrow \theta=π3{}= \tfrac{\pi}{3}
  5. The point (32,π3)\left(\tfrac{3}{2}, \tfrac{\pi}{3}\right)

Total5

  • M
  • A
  • B
How did it go?

Report a problem with this question