MATH 2414 – CALCULUS II Section 10.3 Polar Coordinates Polar coordinate system: Choose a point O in the plane. This will be the pole (origin). Draw a ray starting at O – this is the polar axis (corresponds to the + x-axis in Cartesian coordinates) If P is another point in the plane, let r be the distance from O to P and q (radians) be the angle between the polar axis and line segment OP. P is given by the point (r;q). An angle is + if measured in CCW direction An angle is - if measured in CW direction If P=O, then r=0. (0,q) is the pole for any q If r < 0, points (−r;q) and (r;q) lie on the same line through O and at the same distance jrj from O. NOTE: (−r;q) = (r;q + p) Each point in the polar coordinate system can be written in infinitely many ways. Ex: Plot points: 3p (a) 2; (b) (4;3p) 4 2p 3p (c) −2; (d) 3;− 3 4 Math 2414 Section 10.3 Continued Since a complete CCW rotation is given by 2p, we can write (r;q) as Connection between Polar and Cartesian Coordinate Systems: To convert from polar to rectangular: To convert from rectangular to polar: p Ex: Convert the point 4; to Cartesian coordinates. 6 Ex: Convert the point (−2;2) to polar coordinates. NOTE: When converting from rectangular to polar coordinates, 2 Math 2414 Section 10.3 Continued Polar curves: Ex: What curve is represented by the polar equation r = 3? p Ex: Sketch the polar curve q = 4 Ex: Sketch the polar curve r = 3sinq. Find a cartesian equation for this curve. Circles: r = asinq or r = acosq 3 Math 2414 Section 10.3 Continued Cardioids: r = a ± asinq or r = a ± acosq Ex: Sketch the curve r = 2 − 2sinq. Roses: r = asin(nq) or r = acos(nq). n petals when n is odd (because each petal is traversed twice) 2n petals when n is even a gives the length of the petal 2p Distance between two petals: # of petals To find tip of petal: Solve a = acos(nq) or a = asin(nq) NOTE: The curve passes through the pole (origin) when r = 0. This may be useful when looking at individual petals. Ex: Sketch the curve r = sin(3q). Ex: Sketch r = 3cos(4q). 4 Math 2414 Section 10.3 Continued Tangents to Polar Curves: Sub in f (q) for r so x = r cosq = f (q)cosq and y = r sinq = f (q)sinq dy dy = dq = dx dx dq p Ex: Find the slope of the tangent line to r = sin(3q) at the point q = . 6 5.