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

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.

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.

Conversion Between Polar and Rectangular Polar To Rectangular Rectangular To Polar

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 .

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.

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

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

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) = 0 -----> 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

ARC LENGTH

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

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

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).

Find the area inside the inner loop of

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 15.2695

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