1.5 Polar coordinates

A polar point is written (r,θ), where r is the directed distance from the pole and θ is the angle from the initial line. Cartesian coordinates satisfy x=r cosθ and y=r sinθ.

The same point can have infinitely many representations: (r,θ)=(r,θ+2π) and (−r,θ)=(r,θ+π). Decide whether the problem uses a signed radius or restricts r≥0.

The point (2,π/3) has Cartesian coordinates (1,√3). The representation (−2,4π/3) describes the same point.

Do not treat θ as a length or assume a unique polar pair; angle periodicity and negative r matter.

A polar curve is given by r=f(θ). To understand its shape, identify the interval, zeros of r, symmetries and key angles before plotting points.

If f(−θ)=f(θ), the curve is symmetric about the initial line; if f(π−θ)=f(θ), it is symmetric about the line θ=π/2. Negative r plots in the opposite direction, so sign changes are structural.

For r=2cosθ, r is zero at π/2 and negative for π/2<θ<3π/2; the curve is a circle of diameter 2 on the initial-line side after the negative-radius points are interpreted.

Joining a table of positive radii without checking sign and symmetry can produce the wrong curve.

The area swept by a polar curve from θ=a to θ=b is A=½∫_a^b r² dθ, provided the interval traces the intended region once.

Square r, so a negative radius contributes positive area. Find intersections and choose limits carefully; if the curve crosses the pole or overlaps itself, split the region to avoid counting it twice.

For r=2cosθ from −π/2 to π/2, A=½∫4cos²θ dθ=π, the area of the corresponding circle.

Do not integrate r rather than r², and do not assume the stated interval is automatically one non-overlapping traversal.