Polar Coordinates. Butterflies are among the most celebrated of all insects. Their symmetry can be...
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Transcript of Polar Coordinates. Butterflies are among the most celebrated of all insects. Their symmetry can be...
• Butterflies are among the most celebrated of all insects.
• Their symmetry can be explored with trigonometric functions and a system for plotting points called the polar coordinate system.
The foundation of the polar coordinate system is a horizontal ray that extends to the right. The ray is called the polar axis. The endpoint of the ray is called the pole.
r can be positive, negative, or zero.
Positive angles are measured counterclockwise from the polar axis.
Negative angles are measured clockwise from the polar axis.
• If P is a point with polar coordinates , the rectangular coordinates of P are given by
( , )r
( , )x y
cos
sin
x r
y r
1. To find r, compute the distance from the origin to (x, y).
2. To find , first determine the quadrant that the point lies in.
1
1
1
1
: tan
: tan
: tan
: tan
yI
xy
IIxy
IIIx
yIV
x
2 2 2r x y
Find the polar form of the rectangular point (4, 3).
Find the polar form for the point (-2, 3).
Find the polar coordinates for the rectangular coordinates (-3, 3).
Objective:
The letters x and y represent rectangular coordinates. Write each equation using polar coordinates.
• Transform the following equations from rectangular coordinates to polar coordinates.
2 2
2
2 3 5
5
3 4 2
( 3) 9
x
x y
x
x y
x y
Objective:
The letters represent polar coordinates. Write each equation using rectangular coordinates.
• Transform the following equations from polar coordinates to rectangular coordinates.
3sin
4cos
r
r
• An equation whose variables are polar coordinates is called a polar equation.
• The graph of a polar equation consists of all points whose polar coordinates satisfy the equation.
Graphing a Polar Equation using your Calculator
1. Solve the equation for r in terms of .2. Clear the memory on your calculator.3. Select Mode - Polar. 4. Window ( step determines the number of
points that your calculator will graph)5. Zoom - Standard6. Enter the expression that you found in step 1.
• Let a be a nonzero real number. Then the graph of the equation is a horizontal line a units above the pole if a > 0 and units below the pole if a < 0 .
• The graph of the equation is a vertical line a units to the right of the pole if a > 0 and units to the left of the pole if a < 0.
sinr a a
cosr a
a
• Specific Types of Polar Graphs• Symmetry• Plot each point given in polar coordinates and
find other polar coordinates of the point.