Graphing Polar Coordinates 5015

Angle pos counter clockwise neg clockwise Preliminaries: Pole: Polar Axis: Coordinates: r = θ = origin Positive x-axis Radius (r, θ) Angle pos counter clockwise neg clockwise

Plot the following points: Degree Measure: A(4, 330⁰) B)(3, -90⁰) C(-2, 30⁰) D(-4, -45⁰) D C A B

Radian Measure: A 3, 7𝜋 6 B 2, − 3𝜋 4 C −4, 𝜋 D −5, − 𝜋 3

Naming Polar Coordinates – all four ways (4, 225) 4, −135 −4, −315 −4, 45

Examples: A(5, 150⁰) B(3, -240⁰)

Converting Polar Rectangular

Converting 𝑃(𝑟, 𝜃)→𝑅(𝑥, 𝑦) Remember the unit circle: 𝑐𝑜𝑠𝜃, 𝑠𝑖𝑛𝜃 Now radius can be different values (𝑡, 𝜃) 𝑠𝑖𝑛𝜃= 𝑦 𝑟 cos 𝜃= 𝑥 𝑟 y r 𝜃 𝑦=𝑟𝑠𝑖𝑛 𝜃 𝑟𝑐𝑜𝑠 𝜃=𝑥 x Polar, 𝑃(𝑟,𝜃) rectangular, R(x,y)

TO CONVERT Polar to Rectangular 𝑃(𝑟, 𝜃)→𝑅(𝑥, 𝑦) 𝑃(5, 60°) 𝑦=𝑟𝑠𝑖𝑛 𝜃 𝑥=𝑟𝑐𝑜𝑠 𝜃 𝑦=5 sin 60° 𝑥=5 cos 60° 𝑦= 5 3 2 𝑥= 5 2 𝑅 5 2 , 5 3 2

𝑃 4, 11𝜋 6 𝑥=𝑟𝑐𝑜𝑠 𝜃 𝑦=𝑟𝑠𝑖𝑛 𝜃 𝑥=4 cos 11𝜋 6 𝑦=4 sin 11𝜋 6 𝑥=4∗ 3 2 =2 3 𝑦=4 − 1 2 =−2 𝑅 2 3 ,−2

Convert Rectangular to Polar 𝑅 𝑥, 𝑦 →𝑃(𝑟, 𝜃) + 𝑟= 𝑥 2 + 𝑦 2 𝜃= 𝑡𝑎𝑛 −1 𝑦 𝑥 - If x is negative add 180 or 𝜋 If y only is negative add 2𝜋 𝑅(5, 9) 𝜃= 𝑡𝑎𝑛 −1 9 5 𝑟= 5 2 + 9 2 𝑟= 106 𝜃=61° 𝑃( 106 , 61°)

𝑅(−6, −3) 𝜃= 𝑡𝑎𝑛 −1 𝑦 𝑥 𝑟= 𝑥 2 + 𝑦 2 𝜃= 𝑡𝑎𝑛 −1 −3 −6 𝑟= −6 2 + −3 2 𝑟= 45 𝜃≈26.6 But we know the angle is in the III quadrant we will add 180° 𝑟=3 5 So 𝑃 3 5 ,206.6°

𝑅(−5, 12) 𝑅(3, −4) 𝜃= 𝑡𝑎𝑛 −1 𝑦 𝑥 𝑟= 𝑥 2 + 𝑦 2 𝜃= 𝑡𝑎𝑛 −1 𝑦 𝑥 𝑟= 𝑥 2 + 𝑦 2 𝜃= 𝑡𝑎𝑛 −1 −4 3 𝑟= 3 2 + −4 2 𝜃= 𝑡𝑎𝑛 −1 12 −5 𝑟= −5 2 + 12 2 𝑟=5 𝜃≈−53.1 𝑟=13 𝜃≈−67.4° But we know the angle is in the IV quadrant so we will add 360° We know the angle is in the II quadrant so we add 180° 𝑃 5, 306.9° 𝑃 13, 112.6°