1. By Kevin Dai, Minho Hyun, David Lu
POLAR FUNCTIONS
By Kevin Dai, Minho Hyun, David Lu

2. Polar Intro
Polar coordinates are another way of expressing points in a plane. Instead of being centered at an origin and moving horizontally or vertically, polar coordinates are centered at the pole and measure a radius out from the pole at a given angle.

3. Polar Intro Cont.
Although polar functions are differentiated in r and 𝜽, the coordinates and slope of a line tangent to a polar curve are given in rectangular coordinates.
Therefore you must know a few Polar-Rectangular conversions.

4. Conversion Between Polar and Rectangular
Polar To Rectangular
Rectangular To Polar

5. dy/d𝜃 dx/d𝜃 Slope in Polar Form
The slope formula in polar form of a polar graph is done parametrically and is formed by the derivatives of x = r cos𝜃 and y = r sin𝜃 .

6. Find 𝜃 where the graph of r = 1-sin𝜃 has horizontal tangents.
Example::
Find 𝜃 where the graph of r = 1-sin𝜃 has horizontal tangents.

7. Find θ where the graph of r = 1-sinθ has horizontal tangents.
Step 1: Find dr/dθ
dr/dθ = -cosθ
Step 2: Plug it in the dy/dx equation
(-cosθ(sinθ) + (1 - sinθ)(cosθ))/(-cosθ)(cosθ)-(1-sinθ)(sinθ)
Step 3: Set dy/dx = 0
Step 4: Solve for θ!
θ = pi/2, 3pi/2, pi/6, 5pi/6

8. BUT WE’RE NOT DONE YET!!!!!!!!
*There’s a chance that θ for the vertical and horizontal tangents are equal

9. We also have to find θ of the vertical tangents of r = 1-sinθ
Step 5: set the denominator of dy/dx = 0
(1-sinθ)(-sinθ) + (-cosθ)(cosθ) = > 2sin2θ - sinθ - 1 = 0
Step 6: Solve for θ
θ = 7pi/6, 11pi/6, and pi/2
Step 7: cross out the shared θ
The θ’s of the horizontal tangents to r = 1-sinθ:
θ = 3pi/2, pi/6, 5pi/6

11. ARC LENGTH

12. Find the length of the arc from [0, 2𝝅] for r = 2-2cosθ
Example:
Find the length of the arc from [0, 2𝝅] for r = 2-2cosθ

13. Find the length of the arc from [0, 2pi] for r = 2-2cosθ
Step 1: Find dr/dθ = 2sinθ
Step 2: PLUG IT IN!
L = 16

14. Polar Area
We can also find the area in polar coordinates. F needs to be continuous and non-negative on the interval (a, B) where 0 < B - a < 2π r = f(θ)
We will have to set r=0 so that we can find the bounds.
The formula to the right will give the area of the region bounded by the graph of r = f(θ).

15. Find the area inside the inner loop of

16. Find the area inside the inner loop of
Set the equation equal to 0 so that cosθ = 3/8.
You can then plug into the calculator to find that θ = ± 1.186
These will be your bounds; now all you need to do is plug the equation and the bounds into the formula.
The final answer will be

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18. FRQ Answer Key/Scoring