Lesson 9-1

Example 1
Graph each point.

a.	P(2, 255°)
On a polar plane, sketch the terminal side of a 255° angle in standard position.

	Since r is positive r = 2, find the point on the terminal side of the angle that is 2 units from the pole.

	Notice that point P is on the second circle from the pole.		
		
b.	Q-3.5, 
Sketch the terminal side of an angle measuring  radians in standard position.

	Since r is negative, extend the terminal side of the angle in the opposite direction.  Find the point Q that is 3.5 units from the pole along this extended ray.

	Notice that point Q is halfway between the third and fourth circles from the pole.		

Example 2
Graph R-3, -.

Negative angles are measured clockwise. Sketch the terminal side of an angle of - radians in standard position.

Since r is negative, the point R-3, - is 3 units from the pole along the ray opposite the terminal side that is already drawn.		

Example 3
Name the four different pairs of polar coordinates that represent point S on the graph with the restriction that -360° ≤  ≤ 360°.

One pair of polar coordinates for point S is (2, 300°).

To find another representation, use
(r ,  + 360k°) with k = -1.
(2, 300 + 360°(-1) = (2, -60°)

To find additional polar coordinates, use (-r,  + (2k + 1)180°).
(-2, 300° + (-3)180°) = (-2, -240°)   k = -2

To find a fourth pair, use
(-r,  + (2k + 1)180°) with k = -1.
(-2, 300° + (-1)180°) = (-2, 120°)		

Therefore, (2, 300°), (2, -60°), (-2, -240°), and (-2, 120°) all represent the same point in the polar plane.

Example 4
Graph each polar equation.

a.	r = 2.5
The solutions to r = 2.5 are the ordered pairs (r, ) where r = 2.5 and  is any real number. Some examples are (2.5, 0), 2.5, , and (2.5, ). In other words, the graph of this equation is the set of all ordered pairs of the form (2.5, ). Any point that is 2.5 units from the pole is included in this graph. The graph is the circle centered at the origin with radius 2.5.		
		
b.	 = 
The solutions to this equation are the ordered pairs (r, ) where  =  and r is any real number. Some examples are 3, , -2, , and 1, . In other words, the graph of this equation is the set of all ordered pairs of the form r, . The graph is the line that includes the terminal side of the angle  in standard position.		

Example 5
SURVEYING  A surveyor identifies two landmarks, one 400 feet away 25° to the left and 320 feet away 45° to the right. What is the distance between the two landmarks?

Set up a coordinate system so that turning to the left is a positive angle and turning to the right is a negative angle. With this convention, the landmarks are at L1(400, 25°) and L2(320, -45°).

Now use the Polar Distance Formula.		

P1P2	= 
L1L2	= 	(r1, 1) = (400, 25°), (r2, 2) = (320, -45°)
	 418.1 feet

The landmarks are about 418 feet apart.