Section 10.4 Polar Coordinates and Polar Graphs

Transcription

1 Section 10.4 Polar Coordinates and Polar Graphs

2 Introduction to Polar Curves Parametric equations allowed us a new way to define relations: with two equations. Parametric curves opened up a new world of curves: x y cos sin 4 t t Polar coordinates will introduce a new coordinate system.

3 Introduction to Polar Curves You have only been graphing with standard Cartesian coordinates, which are named for the French philosopher-mathematician, Rene Descartes. Example: Plot ( 3,) y x

4 Polar Coordinates In polar coordinates we identify the origin O as the pole and the positive x-axis as the polar axis. We can then identify each point P in the plane by polar coordinates (r, θ), where r gives the distance from O to P and θ gives the angle from the initial ray to the ray OP. By convention, angles measured in the counterclockwise direction are positive. Since it easier to plot a point by starting with the angle, polar equations are like inverses. θ = independent variable. r = dependent variable. NOTE: The origin O has no well-defined coordinate. For our purposes the coordinates will be (0, θ) for any θ.

5 Example 1 Example: Plot the polar coordinates (3, π 3 ). To plot a point using polar coordinates (r, θ), we often use a polar grid: 3 π 3 First find the angle θ on the polar grid. Now plot the point r units in the direction of the angle.

6 Example Example: Plot the polar coordinates (, 5π 3 ). To plot a point using polar coordinates (r, θ), we often use a polar grid: 5π 3 First find the angle θ on the polar grid. Now plot the point r units in the direction of the angle. If r is negative, the point is plotted r units in the opposite direction.

7 Example 3 Graph the polar curve r = 3 cos θ. Indicate the direction in which it is traced. Notice Polar equations are like inverses. θ = independent variable. r = dependent variable. r 3 cos 0 = 3 3 cos π 6 = cos π 4 =.11 3 cos π 3 = cos π = 0 3 cos π 3 = cos 3π 4 =.1 3 cos 5π 6 =.6 3 cos π = 3 θ 0 π 6 π 4 π 3 π/ π 3 3π 4 5π 6 π

8 The Relationships Between Polar and Cartesian Coordinates Find the relationships between x, y, r, & θ. (r,θ) y Right triangles are always a convenient shape to draw. y r Using Pythagorean Theorem x + y = r x x

9 sin θ The Relationships Between Polar and Cartesian Coordinates Find the relationships between x, y, r, & θ. (r,θ) y x x r 1 cos θ y (cos θ, sin θ) θ What about the angle θ? You can use a reference angle to find a relationship but that would require an extra step. Instead, compare the coordinates to the unit circle coordinates. The red and blue triangles are similar with a scale factor of r = r. Thus 1 x = r cos θ y = r sin θ

10 y = r sin θ The Relationships Between Polar and Cartesian Coordinates Find the relationships between x, y, r, & θ. (r,θ) x r x = r cos θ y θ What about a relationship with x, y, & θ? To find the angle measure θ, it is possible to use the tangent function to find the reference angle. Instead investigate the tangent function and θ: tan θ = sin θ cos θ Therefore: tan θ = y x = r sin θ r cos θ (Remember tangent is also the slope of the radius.)

11 Conversion Between Polar and Cartesian Coordinates When converting between coordinate systems the following relationships are helpful to remember: x = r cos θ y y = r sin θ x x + y = r tan θ = y x NOTE: Because of conterminal angles and negative values of r, there are infinite ways to represent a Cartesian Coordinate in Polar Coordinates.

12 Complete the following: Example 1 a) Convert (1, 3) into polar coordinates. tan 3 1 tan r 1 3 r 4 r b) Express your answer in (a) as many ways as you can.,, n 3 3 4,, 4 n 3 3, 3

13 NOTE: In Cartesian coordinates, every point in the plane has exactly one ordered pair that describes it. Example Find rectangular coordinates for (16,5π/6). x 16cos x y 16sin y ,8

14 Example 3 Use the polar-rectangular conversion formulas to show that the polar graph of r = 4 sin θ is a circle. r 4sin r r 4rsin x y 4y x y y x y y x 4 y A circle centered at (0,) with a radius of units.

15 Since we can easily convert a polar equation into parametric equations, the calculus for a polar equation can be performed with the parametrically defined functions. Conversion Between Polar Equations and Parametric Equations The polar graph of r = f(θ) is the curve defined parametrically by: The slope of x = r cos t = f(t) cos t tangent lines y = r sin t = f(t) sin t is dy/dx not dr/dθ. Example: Write a set of parametric equations for the polar curve r = sin 6θ x r cost y r sin t sin 6tcost sin 6tsin t

16 Example Use polar equation r = sin 3θ to answer the following questions: (a) Find the Cartesian equation of the tangent line at θ = π/6. Parametric Equations: x r cost y r sin t Find dy/dx not dr/dθ: d dy dy/ sin3tsint dx dx/ sin3tcost d 6cos3t sint sin 3t cost 6cos3t costsin 3t sint sin 3tcost sin 3tsin t Find the equation: Find the slope of the tangent line (Remember t = θ): dy dx 3 t y 6 Find the point: x sin 3 6 cos 6 y sin 3 6 sin x 3 1 3

17 Example (Continued) Use polar equation r = sin 3θ to answer the following questions: (b) Find the length of the arc from θ = 0 to θ = π/6. Parametric Equations: x r cost y r sin t Find dy/ and dx/: sin 3tcost sin 3tsin t dy d sin 3tsin t 6cos3t sin t sin 3t cost dx d sin 3tcost 6cos3t cost sin 3t sin t d Use the Arc Length Formula: 6 0 dx dy.7 6cos3 cos sin 3 sin 6cos3 sin sin 3 cos 6 t t t t t t t t 0

18 Example (Continued) Use polar equation r = sin 3θ to answer the following questions: (c) Is the curve concave up or down at θ = π/6. Parametric Equations: Find dy/dx: Find d y/dx : dy dx d y dx dy/ dx/ d x r cost y r sin t d sin 3tcost sin 3tsin t sin3tsint 6cos3 t sin t sin3 t cos t d sin3tcost 6cos3t costsin 3t sint dy dx dx Find value of the second derivative (Remember t = θ): d y dx t 6 4cos 3t5 3cos3t costsin3t sint 40 Since the second derivative is positive, the graph is concave up.

19 Alternate Formula for the Slope of a Tangent Line of a Polar Curve If f is a differentiable function of θ, then the slope of the tangent line to the graph of r = f(θ) at the point (r, θ) is: dy dy/ d f ( )cos( ) f '( )sin dx dx/ d f ( )sin f '( )cos If you do not want to easily convert a polar equation into parametric equations, you can always memorize another formula...

20 Alternate Arc Length Formula for Polar Curves The arc length for a polar curve r(θ) between θ = α and θ = β is given by dr d L r d If you do not want to easily convert a polar equation into parametric equations, you can always memorize another formula...