1.5 Polar coordinates
- Syllabus
- 9231–2028–2029
- Topic
- 1.5
- Level
- AS
Learning objectives
1.5.1Polar coordinates• understand the relations between Cartesian and polar coordinates, and convert equations of curves from Cartesian to polar form and vice versa The convention r 0H will be used.1.5.2Polar curves• sketch simple polar curves, for 02 <G ri or 1 G-rr i or a subset of either of these intervals Detailed plotting of curves will not be required, but sketches will generally be expected to show significant features, such as symmetry, coordinates of intersections with the initial line, the form of the curve at the pole and least/greatest values of r.1.5.3Polar area• recall the formula r d2 1 2 iy for the area of a sector, and use this formula in simple cases.
Convert the locus, not just its symbols
With the syllabus convention $r\ge0$,x=r\cos\theta,\qquad y=r\sin\theta,\qquad r^2=x^2+y^2,\qquad \tan\theta=\frac yx\ (x\ne0).
Cartesian to polar: replace x and y by rcosθ and rsinθ, use x2+y2=r2, then simplify while retaining the stated angle and r≥0 conditions. For x2+y2=4x, r2=4rcosθ, giving r=4cosθ together with the pole r=0; dividing by r without checking would hide that point.
Polar to Cartesian: replace $r\cos\theta$ by x, $r\sin\theta$ by y and $r^2$ by $x^2+y^2$. Forr=2(\cos\theta+\sin\theta),multiplybyrtoobtainx^2+y^2=2x+2y,or(x-1)^2+(y-1)^2=2.
Do not introduce negative-r alternatives: this course uses r≥0. Algebraic conversion can add or lose points when multiplying, squaring or dividing, so check the pole, domain and original equation after simplifying.
Build a polar sketch from angular events and radial extrema
Use the stated interval (normally 0≤θ<2π or −π<θ≤π): test symmetry by replacing theta with −θ, π−θ or θ+π; solve r=0 for pole visits; find intersections with the initial line at allowed angles representing that ray; solve dr/dθ=0 and check endpoints for least/greatest r; then join only the admissible r≥0 branches in increasing theta order.
At a pole value θ0 where r→0, the limiting angle gives the approach direction. Check whether the interval traces into and out of the pole, ends there, or meets it more than once; this distinguishes a crossing, cusp/loop contact or endpoint. Do not plot a formula-produced negative r on the opposite ray under this syllabus convention—restrict to where r is non-negative.
| Feature for r=a(1+cosθ), a>0 | Deduction |
|---|---|
| Symmetry | r(−θ)=r(θ), so symmetry about the initial line |
| Initial line | θ=0 gives (2a,0); the pole is also on every radial line |
| Pole | r=0 at θ=π; both sides approach along the same line, forming the cardioid cusp |
| Radial extrema | 0≤r≤2a; maximum 2a at θ=0, minimum 0 at θ=π |
A polar sketch is not the Cartesian graph of r against theta. The final plane curve must show symmetry, labelled initial-line intersections, correct pole behaviour and least/greatest radial distances; dense plotting is not required.
Choose one traversal before integrating squared radius
A thin sector of angle $d\theta$ has area approximately $\tfrac12r^2d\theta$. Therefore, when $r=f(\theta)$ traces the intended boundary once from $\alpha$ to $\beta$,A=\frac12\int_{\alpha}^{\beta}r^2,d\theta.
Find the boundary angles from intersections, the initial line or r=0. Check the sketch to ensure the interval covers the desired region once. Split at a pole or branch change when necessary. Between two curves on the same rays, use 21∫(router2−rinner2)dθ.
The upper half of $r=a(1+\cos\theta)$ is traced once for $0\le\theta\le\pi$. ThusA=\frac{a^2}{2}\int_0^\pi(1+\cos\theta)^2d\theta=\frac{a^2}{2}\int_0^\pi(1+2\cos\theta+\cos^2\theta)d\theta=\frac{3\pi a^2}{4}.
The integrand is r2, not r, and the factor one-half is essential. Squaring makes area non-negative, but it does not prevent double counting: only the sketch and angular traversal establish correct limits.