G1 - Polar coordinates

G1 - Polar coordinates

A distance and a direction can locate a point just as well as horizontal and vertical coordinates. You will use polar coordinates, explain why their labels are not unique, convert points and equations into Cartesian form, and connect this geometry to complex numbers.

Radius-angle coordinates

Can the pairs (4,π/6)(4,\pi/6) and (−4,7π/6)(-4,7\pi/6) locate the same point? First decide what a radius means. Start at an origin OO, turn through π/6\pi/6 from the positive horizontal direction, then travel 4 units along that ray.

Polar coordinates

The pole is the origin. The initial line is the ray from the pole in the positive horizontal direction. A point has polar coordinates (r,θ)(r,\theta): rr is its distance from the pole and θ\theta is the angle measured anticlockwise from the initial line, in radians. Clockwise angles are negative.

Unless a different convention is explicitly stated, use r≥0r\geq0. The angle interval is part of the description; use the interval given in the question.

This is the same distance-and-direction geometry you met for a complex number. The coordinate order matters: the first entry is a length, the second an angle. Polar (4,π/6)(4,\pi/6) does not mean Cartesian x=4x=4, y=π/6y=\pi/6.

Recall the circle relations: circumference 2πr2\pi r, area πr2\pi r^2, and, for a sector of angle α\alpha in radians, arc length rαr\alpha and area 12r2α\tfrac12r^2\alpha. The sector occupies the fraction α/(2π)\alpha/(2\pi) of the full circle. These familiar length and area relations will help you check polar curves and their areas.

[DIAGRAM: asset_name: Radius and direction in polar coordinates; asset_slug: aqa7367-polar-radius; caption: A positive radius follows its ray and an explicitly signed negative radius reverses it; description: The left panel shows a length four at angle pi over six, reaching Cartesian point (two root three, two), with horizontal and vertical guides. The right panel separately illustrates a signed-radius convention: a radius minus four along angle pi over six reaches the opposite point, equivalently positive radius four at angle minus five pi over six.]
A positive radius follows its ray and an explicitly signed negative radius reverses it

To begin, locate points on the axes without doing any trigonometry.

Worked example 1: a quarter-turn

Question. Describe the position of the point with polar coordinates (5,π/2)(5,\pi/2), and give its Cartesian coordinates.

Plan. Turn anticlockwise through a right angle, then travel the positive radius.

θ=π2 ⟹ positive vertical ray⟹(x,y)=(0,5)\theta=\frac{\pi}{2}\ \Longrightarrow\ \text{positive vertical ray} \qquad\Longrightarrow\qquad (x,y)=(0,5)

Answer. The point is 5 units directly above the pole, with Cartesian coordinates (0,5)(0,5).

Check. Its distance is 02+52=5\sqrt{0^2+5^2}=5, and its direction is a positive quarter-turn.

Change only the sign of the angle. The length stays positive, so the turn changes rather than the direction of travel along the chosen ray.

Worked example 2: a clockwise quarter-turn

Question. Describe the position of the point with polar coordinates (5,−π/2)(5,-\pi/2), and give its Cartesian coordinates.

Plan. Turn clockwise through a right angle, then travel 5 units.

θ=−π2 ⟹ negative vertical ray⟹(x,y)=(0,−5)\theta=-\frac{\pi}{2}\ \Longrightarrow\ \text{negative vertical ray} \qquad\Longrightarrow\qquad (x,y)=(0,-5)

Answer. The point is 5 units directly below the pole, at (0,−5)(0,-5).

Check. Reflecting Worked example 1 in the initial line reverses the vertical coordinate and angle, while preserving distance.

The signs of the two entries play different roles.

Watch out

A negative angle is allowed with a positive radius: it means a clockwise turn. A negative radius needs an explicitly stated signed-radius convention; it is not a negative distance.

Try a different axis direction.

Nonuniqueness and the pole

Turning one full revolution before travelling does not change the point you reach. For r>0r>0, the pairs (r,θ)(r,\theta) and (r,θ+2kπ)(r,\theta+2k\pi) describe the same point for every integer kk.

To choose one angle, use a half-open interval of length 2π2\pi, such as 0≤θ<2π0\leq\theta<2\pi or −π<θ≤π-\pi<\theta\leq\pi. “Half-open” means that exactly one endpoint is included: including both would duplicate the initial direction. For the latter interval the negative horizontal ray uses π\pi, not −π-\pi.

Some questions or software explicitly allow a signed radius. Then r<0r<0 means travelling ∣r∣|r| units opposite to the ray at angle θ\theta. The same point therefore has positive-radius coordinates

(r,θ)=(−r,θ+π)(r<0)(r,\theta)=(-r,\theta+\pi)\qquad(r<0)

after adjusting the new angle into the requested interval. This extends the coordinate convention; it does not make a distance negative.

Worked example 3: choosing one angle

Question. Write (4,7π/6)(4,7\pi/6) with a positive radius and angle in −π<θ≤π-\pi<\theta\leq\pi.

Plan. Keep the radius and subtract a full turn from the angle.

7π6−2π=−5π6\frac{7\pi}{6}-2\pi=-\frac{5\pi}{6}

Answer. (4,−5π/6)(4,-5\pi/6).

Check. The angles differ by 2π2\pi, so their sine and cosine agree. Both rays point left and down.

Now change only the sign of the first entry and explicitly allow signed radii. The point moves to the opposite ray.

Worked example 4: removing a signed negative radius

Question. Under a signed-radius convention, write (−4,7π/6)(-4,7\pi/6) with a positive radius and angle in −π<θ≤π-\pi<\theta\leq\pi.

Plan. Reverse the ray by adding π\pi, then remove a full turn.

rnew=4positive distanceθnew=7π6+π=13π6θnew−2π=π6required interval\begin{aligned} r_{\text{new}}&=4 && \text{positive distance}\\ \theta_{\text{new}}&=\frac{7\pi}{6}+\pi=\frac{13\pi}{6}\\ \theta_{\text{new}}-2\pi&=\frac{\pi}{6} && \text{required interval} \end{aligned}

Answer. (4,π/6)(4,\pi/6). This is the point in the opening question.

Check. Both descriptions give horizontal component 232\sqrt3 and vertical component 22: the two sign reversals cancel.

At the pole, r=0r=0, so every pair (0,θ)(0,\theta) reaches the same point. An assigned coordinate angle is only a label there. The pole has no geometrically determined direction, just as the complex number zero has no argument. Restricting the angle interval does not give the pole a direction.

Watch out

Wrong: (−4,θ)=(4,θ)(-4,\theta)=(4,\theta). With signed radii, change the angle by a half-turn when changing the radius sign. With the usual nonnegative-radius convention, reject a negative radius rather than silently plotting it.

Use the explicitly stated convention in this new case.

Continue this lesson.

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