Polar Coordinates

Differences: Polar vs. Rectangular (0,0) is called the pole Coordinates are in form (r, θ) (0,0) is called the origin Coordinates are in form (x,y)

How to Graph Polar Coordinates Given: (3, л/3)

Answer- STEP ONE Look at r and move that number of circles out Move 3 units out (highlighted in red) 1 2 3

Answer- STEP TWO Look at θ- this tells you the direction/angle of the line Place a point where the r is on that angle. In this case, the angle is л/3 1 2 3

Answer: STEP THREE Draw a line from the origin through the point

Converting Coordinates Remember: The hypotenuse has a length of r. The sides are x and y. By using these properties, we get that: x = rcosθ y=rsinθ tanθ=y/x r2=x2+y2 3, л/3 r y x

CONVERT: Polar to Rectangle: (3, л/3) x=3cos (л/3) x=3cos(60) 1.5 y=3sin(л/3) x=3sin(60) 2.6 New coordinates are (1.5, 2.6) ***x = rcosθ ***y=rsinθ tanθ=y/x r2=x2+y2

CONVERT: Rectangular to Polar: (1, 1) Find Angle: tanθ= y/x tanθ= 1 tan-1(1)= л/4 Find r by using the equation r2=x2+y2 r2=12+12 r= √2 New Coordinates are (√2, л/4) (You could also find r by recognizing this is a 45-45-90 right triangle)

Finding points of intersection Third point does not show up. On r = 1, point is (1, π) On r = 1-2 cos θ, point is (-1, 0)

TANGENTS TO POLAR CURVES To find a tangent line to a polar curve r = f(θ), we regard θ as a parameter and write its parametric equations as: x = r cos θ = f (θ) cos θ y = r sin θ = f (θ) sin θ

Slope of a polar curve Horizontal tangent where dy/dθ = 0 and dx/dθ≠0 Where x = r cos θ = f(θ) cos θ And y = r sin θ = f(θ) sin θ Horizontal tangent where dy/dθ = 0 and dx/dθ≠0 Vertical tangent where dx/dθ = 0 and dy/dθ≠0

TANGENTS TO POLAR CURVES We locate horizontal tangents by finding the points where dy/dθ = 0 (provided that dx/dθ ≠ 0). Likewise, we locate vertical tangents at the points where dx/dθ = 0 (provided that dy/dθ ≠ 0).

TANGENTS TO POLAR CURVES For the cardioid r = 1 + sin θ Find the points on the cardioid where the tangent line is horizontal or vertical.

TANGENTS TO POLAR CURVES Observe that:

TANGENTS TO POLAR CURVES Using Equation 3 with r = 1 + sin θ, we have:

TANGENTS TO POLAR CURVES Hence, there are horizontal tangents at the points (2, π/2), (½, 7π/6), (½, 11π/6) and vertical tangents at (3/2, π/6), (3/2, 5π/6) When θ = 3π/2, both dy/dθ and dx/dθ are 0. So, we must be careful.

Figure 11.32

Length of a Curve in Polar Coordinates Find the length of the arc for r = 2 – 2cosθ sin2A =(1-cos2A)/2 2 sin2A =1-cos2A 2 sin2 (1/2θ) =1-cosθ

Area of region

Find Area of region inside smaller loop

Area of a Surface  

SOURCES (LINKS AND NAMES) 1.aculty.essex.edu/~wang/221/Chap10_Sec3 2. Nate Long 3. Pearson