2Polar CoordinatesUp till now, we have graphed on the Cartesian plane using rectangular coordinatesIn the rectangular coordinate system a point is plotted as (x, y).
3Polar CoordinateIn a polar coordinate system, we select a pint, called to pole, and then a ray with vertex at the pole, called the polar axis.We still use an ordered pair to graph.The new ordered pair is (r, θ).If r > 0, then r is the distance of the point from the pole (like the origin)
4Polar Coordinatesθ is the angle (in degrees or radians) formed by the polar axis and a ray from the pole.We call the ordered pair (r, θ) the polar coordinates of the point.
5Polar CoordinatesSince angles have several different ways to name them, there are an infinite number of polar coordinates for each point. (Unlike rectangular coordinates which have only one name for point on the Cartesian plane.)
6Polar Coordinates Find four names for the point We are given a positive radius and a positive angle. We want to find a positive angle and a negative r, a pos r and neg angle, and a neg angle and neg r.
7Steps for finding other polar coordinates 1. Subtract 360o (or 2p) to get a negative angle2. Add 180o (or p) to change the r to negative (half-way around the circle to be on the other side of the polar graph)3. Add or subtract 360o (or p) to find the other angle
12Polar to Rectangular Coordinates You can check your answers using your calculator.First do 2nd APPSChoosePut in polar coordinatesHit enter
13Steps for Converting from Rectangular to Polar Coordinates 1. Always plot the point (x, y) first2. To find r, r2 = x2 + y2 (Look familiar?)3. To find q, remember that we only know x and y. Therefore, the trig value that we can use involves only x and y – tangent.
14Converting from Rectangular Coordinates to Polar Coordinates Find polar coordinates of a point whose rectangular coordinates area. (0, 3)b. (2, -2)c. (-3, 3)
15Transforming an Equation from Polar to Rectangular Form Transform the equation r = cos q.We do not have a formula for justcos q, but we do have one for r cos q.Multiply both sides by r to get r cos q.That gives us r2 = r cos q
16Transforming an Equation from Polar to Rectangular Form r2 = x2 + y2 and r cos q = xSo, x2 + y2 = xThis is the equation of a circleFind the answer by completing the square
17Transforming an Equation from Polar to Rectangular Form (x2 – x ) + y2 = 0 (complete the square)(x2 – x + ¼) + y2 = ¼(x - ½)2 + y2 = ¼This is a circle whose center is (½, 0) and whose radius is ½
18Transforming an Equation from Rectangular to Polar Form
19Transforming from Rectangular to Polar Form y2 = 2x(r sin q)2 = 2 r cos qr2 sin2 q = 2r cos q