You are given the polar curve r = cos(θ) + sin(θ)

a) list all of the points (r,θ) where the tangent line is horizontal. in entering your answer, list the points starting with the smallest value of and limit yourself to (r > or equal to 0) and θ: [0, 2pi] if two or more points share the same value of r, list those starting with the smallest value of θ.

point 1 (r,θ): ?
point 2 (r,θ): ?
point 3 (r,θ): ?

b) repeat this with a tangent vertical line

  a) 1.30656∠(3π/8), 0.541196∠(15π/8)

  b) 1.30656∠(π/8), 0.541196∠(5π/8)

Step-by-step explanation:

The critical points can be found in polar coordinates by considering ...

We can simplify the effort a little bit by rewriting r as …

Then, filling in function and derivative values, we have …

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(a) For horizontal tangents, dy/dx = 0, so we have …

We can use reference angles for the “r” expressions and write the two horizontal tangent point (r, θ) values of interest as …

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(b) For vertical tangents, dy/dx = undefined, so we have …

Again using reference angles for “r”, the two vertical tangent point values of interest are …

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The attached graph shows the angle values in degrees and the radius values as numbers. The points of tangency are mirror images of each other across the line y=x.

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