Polar Coordinates Def: The polar coordinate system uses distance and directions to specify the location of a point in the plane. To set up this system, we choose a fixed point O in the plane called the pole (or origin) and draw from O a ray (half‐line) called polar axis. Then each point P can be assigned polar coordinates P(r, θ ) where r is the distance from O to P. θ is the angle between polar axis and the segment OP . 3
π
π
Exp: Plot the points (a) (1, π ) (b) (3,− ) (c) ( −4, ) 4
6
4
Sol: (a) (b) (c) 4
2
Exercise: (1) ( −2, π ) (2) (3,− π ) 3
3
Relationship between Polar and Rectangular coordinates. 1. x = r cos θ and y = r sin θ y
2. r 2 = x 2 + y 2 and tan θ = ( x ≠ 0) x
Exp: 2
1. Find the rectangular for the point that has polar coordinates (4, π ) . 3
2
1
Sol: x = r cos θ = 4 ⋅ cos π = 4(− ) = −2 3
2
2
3
=2 3 y = r sin θ = 4 ⋅ sin π = 4 ⋅
3
2
P( x, y ) = (−2,2 3 ) 2.Find polar coordinates for the point that has rectangular coordinates (2,−2) . 2
2
2
Sol: r = x + y = 8 r = ±2 2 y
π
3
tan θ = x = −1 θ = 4 π or − 4 π
3
Since (2,−2) ∈ quadrant IV , ( 2 2 ,− 4 ) ( −2 2 ,− 4 π ) . Polar Equation r = f (θ ) Exp: Express the equation x 2 = 4 y in polar coordinates. Sol: x 2 = 4 y sin θ
2
(
r
cos
θ
)
=
4
r
sin
θ
⇒
r
=
4
= 4 tan θ sec θ
cos 2 θ
Exp: Express the polar equation in rectangular coordinates. 1. r = 5 sec θ Sol: r = 5
1
cos θ
⇒ r cos θ = 5 ⇒ x = 5
2. r = 2 sin θ Sol: r 2 = 2r sin θ ⇒ x 2 + y 2 = 2 y ⇒ x 2 + ( y − 1) 2 = 1 Exercise: Convert the equation to polar form 1. x 2 + y 2 = 9 ⇒ r 2 = 9, r = ±3 2. x = y ⇒ r cos θ = r sin θ ⇒ tan θ = 1 ⇒ θ =
π
4 Convert the polar equation to rectangular coordinates. 1. r = 7 ⇒ r 2 = 49 ⇒ x 2 + y 2 = 49 2. r cos θ = 6 ⇒ x = 6 Graphs of Polar Equation Exp: r = 3 Sketch the graph. Sol: r 2 = 9 ⇒ x 2 + y 2 = 9 Exp: θ =
π
.
4 y
Sol: tan θ = 1 = 1 ⇒ x = y
x
Exp: r = 2 sin θ Sketch the graph. Sol: r 2 = 2r sin θ ⇒ x 2 + y 2 = 2 y ⇒ x 2 + ( y − 1) 2 = 1 Exp: Sketch the graph r = 2 + 2 cos θ Sol: Exp: Sketch the graph curve r = cos 2θ . Sol: The graph of r = cos 2θ sketched in rectangular coordinates. The graph of r = cos 2θ sketched in polar coordinates. Exercise: Sketch the graph 1. r = 3 2. r = 3 cos θ 2
r = 3r cos θ
x 2 + y 2 = 3x
3
3
( x- ) 2 + y 2 = ( ) 2
2
2 3. r = 1 + 2 cos θ 4. r = sin 3θ