Section 8.2 Graphs of Polar Equations
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1 Section 8. Graphs of Polar Equations Graphing Polar Equations The graph of a polar equation r = f(θ), or more generally F(r,θ) = 0, consists of all points P that have at least one polar representation (r, θ) whose coordinates satisfy the equation. EXAMPLE: Sketch the polar curve θ =. Solution: This curve consists of all points (r,θ) such that the polar angle θ is radian. It is the straight line that passes through O and makes an angle of radian with the polar axis. Notice that the points (r,) on the line with r > 0 are in the first quadrant, whereas those with r < 0 are in the third quadrant. EXAMPLE: Sketch the following curves: (a) r =, 0 θ π. (b) r = θ, 0 θ 4π. (c) r = cosθ, 0 θ π.
2 EXAMPLE: Sketch the curve r =, 0 θ π. Solution : Since r =, it follows that r = 4. But r = x +y, therefore x +y = 4 which is a circle of radius with the center at the origin. Solution : We have r, theta Pi 6 r, theta Pi 6 r, theta Pi 6 r, theta 4 Pi 6 r, theta Pi 6 r, theta 6 Pi 6 r, theta 7 Pi 6 r, theta 8 Pi 6 r, theta 9 Pi 6 r, theta 0 Pi 6 r, theta Pi 6 r, theta Pi 6
3 EXAMPLE: Sketch the curve r = θ, 0 θ 4π. r theta, theta Pi r theta, theta Pi r theta, theta Pi r theta, theta 4 Pi r theta, theta Pi r theta, theta 6 Pi r theta, theta 7 Pi r theta, theta 8 Pi r theta, theta 9 Pi r theta, theta 0 Pi r theta, theta Pi r theta, theta Pi
4 EXAMPLE: Sketch the curve r = cosθ, 0 θ π. Solution : Since r = cosθ, it follows that r = rcosθ. But r = x + y and rcosθ = x, therefore x +y = x. This can be rewritten as (x ) +y = which is a circle of radius with the center at (,0). Solution : We have r cos theta, theta Pi r cos theta, theta Pi r cos theta, theta Pi r cos theta, theta 4 Pi r cos theta, theta Pi r cos theta, theta 6 Pi r cos theta, theta 7 Pi r cos theta, theta 8 Pi r cos theta, theta 9 Pi r cos theta, theta 0 Pi r cos theta, theta Pi r cos theta, theta Pi
5 EXAMPLES:
6 EXAMPLE: Sketch the curve r = +sinθ, 0 θ π (cardioid). r sin theta, theta Pi 6.0 r sin theta, theta Pi 6.0 r sin theta, theta Pi r sin theta, theta 4 Pi 6.0 r sin theta, theta Pi 6.0 r sin theta, theta 6 Pi r sin theta, theta 7 Pi 6.0 r sin theta, theta 8 Pi 6.0 r sin theta, theta 9 Pi r sin theta, theta 0 Pi 6.0 r sin theta, theta Pi 6.0 r sin theta, theta Pi
7 EXAMPLE: Sketch the curve r = cosθ, 0 θ π (cardioid). r cos theta, theta Pi 6. r cos theta, theta Pi 6. r cos theta, theta Pi r cos theta, theta 4 Pi 6. r cos theta, theta Pi 6. r cos theta, theta 6 Pi r cos theta, theta 7 Pi 6. r cos theta, theta 8 Pi 6. r cos theta, theta 9 Pi r cos theta, theta 0 Pi 6. r cos theta, theta Pi 6. r cos theta, theta Pi
8 EXAMPLE: Sketch the curve r = +4cosθ, 0 θ π. r 4cos theta, theta Pi 6 r 4cos theta, theta Pi 6 r 4cos theta, theta Pi r 4cos theta, theta 4 Pi 6 r 4cos theta, theta Pi 6 r 4cos theta, theta 6 Pi r 4cos theta, theta 7 Pi 6 r 4cos theta, theta 8 Pi 6 r 4cos theta, theta 9 Pi r 4cos theta, theta 0 Pi 6 r 4cos theta, theta Pi 6 r 4cos theta, theta Pi
9 EXAMPLE: Sketch the curve r = cos(θ), 0 θ π (four-leaved rose). r cos theta, theta Pi 6 r cos theta, theta Pi 6 r cos theta, theta Pi 6 r cos theta, theta 4 Pi 6 r cos theta, theta Pi 6 r cos theta, theta 6 Pi 6 r cos theta, theta 7 Pi 6 r cos theta, theta 8 Pi 6 r cos theta, theta 9 Pi 6 r cos theta, theta 0 Pi 6 r cos theta, theta Pi 6 r cos theta, theta Pi 6 9
10 EXAMPLE: Sketch the curve r = sin(θ), 0 θ π (four-leaved rose). r sin theta, theta Pi 6 r sin theta, theta Pi 6 r sin theta, theta Pi 6 r sin theta, theta 4 Pi 6 r sin theta, theta Pi 6 r sin theta, theta 6 Pi 6 r sin theta, theta 7 Pi 6 r sin theta, theta 8 Pi 6 r sin theta, theta 9 Pi 6 r sin theta, theta 0 Pi 6 r sin theta, theta Pi 6 r sin theta, theta Pi 6 0
11 EXAMPLE: Sketch the curve r = sin(θ), 0 θ π (three-leaved rose). r sin theta, theta Pi r sin theta, theta Pi r sin theta, theta Pi r sin theta, theta 4 Pi r sin theta, theta Pi r sin theta, theta 6 Pi r sin theta, theta 7 Pi r sin theta, theta 8 Pi r sin theta, theta 9 Pi r sin theta, theta 0 Pi r sin theta, theta Pi r sin theta, theta Pi
12 EXAMPLE: Sketch the curve r = sin(4θ), 0 θ π (eight-leaved rose). r sin 4theta, theta Pi 6 r sin 4theta, theta Pi 6 r sin 4theta, theta Pi 6 r sin 4theta, theta 4 Pi 6 r sin 4theta, theta Pi 6 r sin 4theta, theta 6 Pi 6 r sin 4theta, theta 7 Pi 6 r sin 4theta, theta 8 Pi 6 r sin 4theta, theta 9 Pi 6 r sin 4theta, theta 0 Pi 6 r sin 4theta, theta Pi 6 r sin 4theta, theta Pi 6
13 EXAMPLE: Sketch the curve r = sin(θ), 0 θ π (five-leaved rose). r sin theta, theta Pi 6 r sin theta, theta Pi 6 r sin theta, theta Pi 6 r sin theta, theta 4 Pi 6 r sin theta, theta Pi 6 r sin theta, theta 6 Pi 6 r sin theta, theta 7 Pi 6 r sin theta, theta 8 Pi 6 r sin theta, theta 9 Pi 6 r sin theta, theta 0 Pi 6 r sin theta, theta Pi 6 r sin theta, theta Pi 6
14 EXAMPLE: Sketch the curve r = sin(6θ), 0 θ π (twelve-leaved rose). r sin 6theta, theta Pi 6 r sin 6theta, theta Pi 6 r sin 6theta, theta Pi 6 r sin 6theta, theta 4 Pi 6 r sin 6theta, theta Pi 6 r sin 6theta, theta 6 Pi 6 r sin 6theta, theta 7 Pi 6 r sin 6theta, theta 8 Pi 6 r sin 6theta, theta 9 Pi 6 r sin 6theta, theta 0 Pi 6 r sin 6theta, theta Pi 6 r sin 6theta, theta Pi 6 4
15 EXAMPLE: Sketch the curve r = sin(7θ), 0 θ π (seven-leaved rose). r sin 7theta, theta Pi 6 r sin 7theta, theta Pi 6 r sin 7theta, theta Pi 6 r sin 7theta, theta 4 Pi 6 r sin 7theta, theta Pi 6 r sin 7theta, theta 6 Pi 6 r sin 7theta, theta 7 Pi 6 r sin 7theta, theta 8 Pi 6 r sin 7theta, theta 9 Pi 6 r sin 7theta, theta 0 Pi 6 r sin 7theta, theta Pi 6 r sin 7theta, theta Pi 6
16 EXAMPLE: Sketch the curve r = + sin(0θ), 0 θ π. 0 r sin 0theta 0, theta Pi 6 r sin 0theta 0, theta Pi 6 r sin 0theta 0, theta Pi 6 r sin 0theta 0, theta 4 Pi 6 r sin 0theta 0, theta Pi 6 r sin 0theta 0, theta 6 Pi 6 r sin 0theta 0, theta 7 Pi 6 r sin 0theta 0, theta 8 Pi 6 r sin 0theta 0, theta 9 Pi 6 r sin 0theta 0, theta 0 Pi 6 r sin 0theta 0, theta Pi 6 r sin 0theta 0, theta Pi 6 6
17 EXAMPLE: Match the polar equations with the graphs labeled I-VI: (a) r = sin(θ/) (c) r = sinθ+sin (θ/) (e) r = +4cos(θ) (b) r = sin(θ/4) (d) r = θsinθ (f) r = / θ 7
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