Some types of POLAR CURVES

Sketch these graphs: 𝑦= sin 𝑥 𝑦= cosec 𝑥 𝑦= cos 𝑥 𝑦= sec 𝑥 Polar curves KUS objectives BAT sketch curves based on their Polar equations 𝑥=𝑟𝑐𝑜𝑠𝜃 𝑦=𝑟𝑠𝑖𝑛𝜃 𝜃=𝑎𝑟𝑐𝑡𝑎𝑛 𝑦 𝑥 𝑟 2 = 𝑥 2 + 𝑦 2 Starter: Sketch these graphs: 𝑦= sin 𝑥 𝑦= cosec 𝑥 𝑦= cos 𝑥 𝑦= sec 𝑥 𝑦= tan 𝑥 𝑦= cot 𝑥

We ignore situations where r < 0 Notes 1 𝑥=𝑟𝑐𝑜𝑠𝜃 𝑦=𝑟𝑠𝑖𝑛𝜃 𝜃=𝑎𝑟𝑐𝑡𝑎𝑛 𝑦 𝑥 𝑟 2 = 𝑥 2 + 𝑦 2 We do not plot any points for a polar curve that give a negative value of r. If you think about it, if you get a negative value for r, the logical way to deal with it would be to plot it in the opposite direction However, changing the direction would mean that the angle used to calculate the value is now different, so the pair of values cannot go together We ignore situations where r < 0 http://mathworld.wolfram.com/topics/PolarCurves.html

𝑟=2+ 𝑐𝑜s 𝜃 −𝜋 ≤𝜃≤𝜋 WB12a Cartesian to Polar coordinates I Table Use Radians WB12a Cartesian to Polar coordinates I a) CARDIOID 𝑟=2+ 𝑐𝑜s 𝜃 −𝜋 ≤𝜃≤𝜋 for θ − 5 6 𝜋 − 2 3 𝜋 − 1 2 𝜋 − 1 3 𝜋 − 1 6 𝜋 r 1 6 𝜋 1 3 𝜋 1 2 𝜋 2 3 𝜋 5 6 𝜋 𝜋 1.134 1.5 2 2.5 2.866 3 Table 2.866 2.5 2 1.5 1.134 1

𝑟 2 =16 𝑐𝑜s 2𝜃 −𝜋 ≤𝜃≤𝜋 WB12b Cartesian to Polar coordinates I Table Use Radians WB12b Cartesian to Polar coordinates I b) LEMINISCATE 𝑟 2 =16 𝑐𝑜s 2𝜃 −𝜋 ≤𝜃≤𝜋 for θ − 5 6 𝜋 − 2 3 𝜋 − 1 2 𝜋 − 1 3 𝜋 − 1 6 𝜋 r 1 6 𝜋 1 3 𝜋 1 2 𝜋 2 3 𝜋 5 6 𝜋 𝜋 Table

𝑟=2 𝑒 𝜃 7 −𝜋 ≤𝜃≤𝜋 WB12c Cartesian to Polar coordinates I Table Use Radians WB12c Cartesian to Polar coordinates I c) SPIRAL 𝑟=2 𝑒 𝜃 7 −𝜋 ≤𝜃≤𝜋 for θ − 5 6 𝜋 − 2 3 𝜋 − 1 2 𝜋 − 1 3 𝜋 − 1 6 𝜋 r 1 6 𝜋 1 3 𝜋 1 2 𝜋 2 3 𝜋 5 6 𝜋 𝜋 Table

𝑟=4 sin 3𝜃 −𝜋 ≤𝜃≤𝜋 WB12d Cartesian to Polar coordinates I Table Use Radians WB12d Cartesian to Polar coordinates I d) ROSE 𝑟=4 sin 3𝜃 −𝜋 ≤𝜃≤𝜋 for θ − 5 6 𝜋 − 2 3 𝜋 − 1 2 𝜋 − 1 3 𝜋 − 1 6 𝜋 r 1 6 𝜋 1 3 𝜋 1 2 𝜋 2 3 𝜋 5 6 𝜋 𝜋 Table

𝑟=2 𝑐𝑜𝑠 𝜃 𝑟=2 𝑠𝑖𝑛 𝜃 𝑟=2− sin 𝜃 𝑟=2− sin 2𝜃 𝑟=2− sin 3𝜃 Use Radians WB13 Sketch the curves with these Polar equations 𝑟=2 𝑐𝑜𝑠 𝜃 𝑟=2 𝑠𝑖𝑛 𝜃 𝑟=2− cos 𝜃− 1 4 𝜋 𝑟=2− sin 𝜃 𝑟=2− sin 2𝜃 𝑟=2− sin 3𝜃

Skills TASKS: SK103 sketch polar curves SK105 curves match1 𝑥=𝑟𝑐𝑜𝑠𝜃 𝑦=𝑟𝑠𝑖𝑛𝜃 𝜃=𝑎𝑟𝑐𝑡𝑎𝑛 𝑦 𝑥 𝑟 2 = 𝑥 2 + 𝑦 2 Skills TASKS: SK103 sketch polar curves SK105 curves match1 SK106 curves match2 SK107 curves to equations Skills Exercise : SK104 polar curves

Crucial points Summary I (r, θ) origin Initial line Make sure that you can covert between cartesian and polar coordinates. cartesian (x, y) polar (r, θ) θ is In the range [-, ] or [0, 2] 2. Remember that cos θ = cos (- θ) If the equation only involves cos it will be symmetrical in the initial line 3. We can use periodicity of the functions to help us e.g. tan has a period of . Therefore r = tan 5θ will repeat itself after an angle of

One thing to improve is – KUS objectives BAT sketch curves based on their Polar equations self-assess One thing learned is – One thing to improve is –

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