1. Polar Coordinates

2. 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)

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

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

5. 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

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

7. 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

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

9. 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 right triangle)

10. 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)

11. 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

12. 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

13. 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).

14. 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.

15. TANGENTS TO POLAR CURVES
Observe that:

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

17. 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.

18. Figure 11.32

19. 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

20. Area of region

21. Find Area of region inside smaller loop

24. Area of a Surface

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