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Home , Polar coordinate system

AP BC Section 10.4: POLAR COORDINATES AND POLAR GRAPHS, pg. 729

POLAR COORDINATES

To form the polar in the , fix a O, called the pole (or ), and construct from O an initial ray called the polar axis, as shown in the figure.

Then each point P in the plane can be assigned polar coordinates r, , as follows.

rOP directed from to

  directed , counter clockwise from polar axis to OP

With rectangular coordinates, each point  x, y has a unique representation. This is not true with polar coordinates. For instance, the coordinates r,  and r,2   represent the same point. Also because r is a directed distance, the coordinates r,  and r,   represent the same point. AP CALCULUS BC Section 10.4: POLAR COORDINATES AND POLAR GRAPHS, pg. 729

COORDINATES CONVERSION

To establish the relationship between polar and rectangular coordinates , let the polar axis to coincide with the positive x-axis and the pole with the origin.

y tan  x

x y cos  , and sin  r r

AP CALCULUS BC Section 10.4: POLAR COORDINATES AND POLAR GRAPHS, pg. 729

Sample Problem #1:

Convert each of the following points into the given coordinate system.

2 a) 4, into Cartesian coordinates 3 b) 1, 1 into Polar coordinates

AP CALCULUS BC Section 10.4: POLAR COORDINATES AND POLAR GRAPHS, pg. 729

Sample Problem #2:

Convert each of the following into an equation in the given coordinate system.

3 a) Convert 25x xxy 1 into polar coordinates.

b) Convert r 8cos into Cartesian coordinates.

AP CALCULUS BC Section 10.4: POLAR COORDINATES AND POLAR GRAPHS, pg. 729

COMMON POLAR GRAPHS

LINES: Some lines have fairly simple equations in polar coordinates.

1.   

2. racos 

3. rbsin 

AP CALCULUS BC Section 10.4: POLAR COORDINATES AND POLAR GRAPHS, pg. 729

Sample Problem #3: USE YOUR POLAR COORDINATE GRAPH PAPER

AP CALCULUS BC Section 10.4: POLAR COORDINATES AND POLAR GRAPHS, pg. 729

CIRCLES:

AP CALCULUS BC Section 10.4: POLAR COORDINATES AND POLAR GRAPHS, pg. 729

Sample Problem #4: USE YOUR POLAR COORDINATE GRAPH PAPER

AP CALCULUS BC Section 10.4: POLAR COORDINATES AND POLAR GRAPHS, pg. 729

CARDIOIDS and LIMACONS:

Sample Problem #5: USE YOUR POLAR COORDINATE GRAPH PAPER

AP CALCULUS BC Section 10.4: POLAR COORDINATES AND POLAR GRAPHS, pg. 729

TANGENTS WITH POLAR COORDINATES

AP CALCULUS BC Section 10.4: POLAR COORDINATES AND POLAR GRAPHS, pg. 729

Sample Problem #5:

AP CALCULUS BC Section 10.4: POLAR COORDINATES AND POLAR GRAPHS, pg. 729

AREA WITH POLAR COORDINATES

AP CALCULUS BC Section 10.4: POLAR COORDINATES AND POLAR GRAPHS, pg. 729

Sample Problem #6:

AP CALCULUS BC Section 10.4: POLAR COORDINATES AND POLAR GRAPHS, pg. 729

Sample Problem #7:

Sample Problem #8:

AP CALCULUS BC Section 10.4: POLAR COORDINATES AND POLAR GRAPHS, pg. 729

Sample Problem #9:

AP CALCULUS BC Section 10.4: POLAR COORDINATES AND POLAR GRAPHS, pg. 729

ARC LENGTH WITH POLAR COORDINATES

The arc length formula for polar coordinates is:

Sample Problem #10:

AP CALCULUS BC Section 10.4: POLAR COORDINATES AND POLAR GRAPHS, pg. 729

SURFACE WITH POLAR COORDINATES

AP CALCULUS BC Section 10.4: POLAR COORDINATES AND POLAR GRAPHS, pg. 729

HOMEWORK:

SECTION 1:

AP CALCULUS BC Section 10.4: POLAR COORDINATES AND POLAR GRAPHS, pg. 729

AP CALCULUS BC Section 10.4: POLAR COORDINATES AND POLAR GRAPHS, pg. 729

SECTION 2:

AP CALCULUS BC Section 10.4: POLAR COORDINATES AND POLAR GRAPHS, pg. 729