University:
University of Greenwich
Course:
BSC152 | Engineering Physics
Academic year:

2024
Views:

368

Pages:

4
Author:

Umar Syed

Complex numbers Z = a + bi Z = r(cosΘ + isinΘ) a = rcosΘ b = rsinΘ (0, -3) -> 0 - 3i a = rcosΘ = 0 Θ = 3π/2 3(cos(3π/2) + isin(3π/2)) Polar Graphs Discovery AP PreCalculus Name ________ Sketch each of the following graphs, noting any intercepts on the theta = 0, theta = pi/2, theta = pi, and theta = 3pi/2 rays. You should also note the first point on the graph. You may use your calculator, but watch AS IT graphs, so you can see the direction the graph is moving. sqrt(2) approx 1.4 sqrt(3) approx 1.7 3sqrt(2) / 2 approx 2.1 r = 2 + 2 sin(theta) r = 2 + 2 cos(theta) r = 2 - 2 sin(theta) r = 2 - 2 cos(theta) r = 3 + cos(theta) r = 3 + sin(theta) r = 3 - cos(theta) r = 3 - sin(theta) r = 2 + cos(theta) r = 1 + 3 cos(theta) r = 1 + 3 sin(theta) r = 1 - 3 cos(theta) r = 1 - 3 sin(theta) r = 1 + 2 cos(theta) r = 1 - 2 sin(theta) r = 4 cos(2 theta) r = 4 cos(3 theta) r = 3 cos(4 theta) r = 3 cos(5 theta) r = 4 sin(2 theta) r = 2 cos(3 theta) (x-2)^2 + (y+3)^2 = 25 x = 2 + 5 cos(theta) y = -3 + 5 sin(theta) r = 2 sin(3 theta) r = 4 sin(5 theta) I. Limacons: Cardiods, with/without Loops r = +/- a +/- b cos(theta) r = +/- a +/- b sin(theta) Look at the relationship of a/b for each of the following types of limacons: Cardioid (#1-4): Limacon with a loop (#10-15): Limacon without a loop or Convex (#5-9): How does r = a + b cos(theta) differ from r = a + b sin(theta)? horizontal (arrow pointing down) vertical (arrow pointing down) How does r = a + b cos(theta) differ from r = a - b cos(theta)? bigger on right (arrow pointing down) bigger on left (arrow pointing down) II. Rose Curves: r = +/- a cos(n theta) r = +/- a sin(n theta) Where do the petals for r = a cos(n theta) lie (2 answers)? 0 + 2pi/n or 0 + pi/n Where do the petals for r = a sin(n theta) lie (2 answers)? pi/(2n) + 2pi/n or pi/(2n) + pi/n How can you tell how long the leaves are (maximum r-value)? a Study "n" in all of the equations, how can this number tell you how many leaves there will be? if n is even, there are 2n petals period is pi if n is odd, there are n petals period is 2pi