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Handouts for Section 11.3: Polar Coordinates • a Coordinate System

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Handouts for Section 11.3: Polar Coordinates • a Coordinate System Handouts for Section 11.3: Polar coordinates • A coordinate system represents a point in the plane by an ordered pair of num- bers called coordinates. Cartesian coordinate system: We choose two perpendicular axes, usually a hori- zontal line drawn to the right and a vertical line drawn up. If P is any point in the plane, let x be its directed distance from the vertical axis and let y be its directed distance from the horizontal axis. The point P is represented by the ordered pair (x, y) and x, y are Cartesian coordinates of P . Polar coordinate system: We choose a point O in the plane called the pole and draw a ray (half-line) starting at O called the polar axis. The polar axis is usually horizontal and drawn to the right. If P is any point in the plane, let r be the dis- tance from O to P and let θ be the angle between the polar axis and the line OP (θ is usually measured in radians). The point P is represented by the ordered pair (r, θ) and r, θ are polar coordinates of P . y−axis r P(r, O ) y P(x,y) O x−axis Ploar axis x Pole Ploar Coordinate System Cartesian Coordinate System Extension: If r< 0, then the pair (r, θ) represents the point P on the directed line that passes through the pole and forms an angle θ with the polar axis, at distance |r| to the left from the pole. O Ploar axis r P(r, O ) 1 A point can be represented in different ways in the polar coordinate system. For example, (r, θ), (r, θ +2π), (−r, θ + π) represent the same point. r P(r, O )=P(−r,O+ π )=P(r, O+2 π ) O+π O Ploar axis • If the pole and polar axis of the polar coordinate system coincide with the origin and the positive x-axis of the Cartesian coordinate system, and if the point P has Cartesian coordinates (x,y) and polar coordinates (r, θ), then x = r cos θ, y = r sin θ Thus, 2 2 2 y r = x + y , tan θ = x • Polar curves: The graph of a polar equation r = f(θ) or more generally F (r, θ)=0 is the set of all points P with at least one polar representation (r, θ) whose coordinates satisfy the given equation. Examples: (1) The curve represented by the polar equation r = 3 is the circle centered at the pole O with radius 3. In general, the polar equation r = c where c is a given real number, represents the circle centered at the pole O with radius |c|. (2) The curve represented by the polar equation θ = π/3 is the line that passes through the pole O and makes an angle of π/3 with the polar axis. In general, the polar equation θ = c (radians), where c is a given real number, represents the line that passes through the pole O and makes an angle of c radians with the polar axis. • Tangents to polar curves: The polar curve r = f(θ) can be represented in the Cartesian coordinate system by the parametric equations x = f(θ) cos θ, y = f(θ) sin θ The slope of the tangent can be then computed using the formula dy dy dθ = dx dx dθ 2.
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