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Pre-Calculus For Dummies, 2nd Edition By Yang Kuang, Elleyne Kase Polar coordinates are an extremely useful addition to your mathematics toolkit because they allow you to solve problems that would be extremely ugly if you were to rely on standard x- and ycoordinates. In order to fully grasp how to plot polar coordinates, you need to see what a polar coordinate plane looks like.

A blank polar coordinate plane (not a dartboard).

In the figure, you can see that the plane is no longer a grid of rectangular coordinates; instead, it’s a series of concentric circles around a central point, called the pole. The plane appears this way because the polar coordinates are a given radius and a given angle in standard position from the pole. Each circle represents one radius unit, and each line represents the special angles from the unit circle.

Because you write all points on the polar plane as

in order to graph a point on the polar plane, you should find theta first and then locate r on that line. This approach allows you to narrow the location of a point to somewhere on one of the lines representing the angle. From there, you can simply count out from the pole the radial distance. If you go the other way and start with r, you may find yourself in a pickle when the problems get more complicated. For example, to plot point E at

which has a positive value for both the radius and the angle — you simply move from the pole counterclockwise until you reach the appropriate angle (theta). You start there in the following list: 1. Locate the angle on the polar coordinate plane. Refer to the figure to find the angle:

2. Determine where the radius intersects the angle. Because the radius is 2 (r = 2), you start at the pole and move out 2 spots in the direction of the angle. 3. Plot the given point. At the intersection of the radius and the angle on the polar coordinate plane, plot a dot and call it a day! This figure shows point E on the plane.

Visualizing simple and complex polar coordinates.

Polar coordinate pairs can have positive angles or negative angles for values of theta. In addition, they can have positive and negative radii. This concept is new; in past classes you’ve always heard that a radius must be positive. When graphing polar coordinates, though, the radius can be negative, which means that you move in the opposite direction of the angle from the pole.

Because polar coordinates are based on angles, unlike Cartesian coordinates, polar coordinates have many different ordered pairs. Because infinitely many values of theta have the same angle in standard position, an infinite number of coordinate pairs describe the

same point. Also, a positive and a negative co-terminal angle can describe the same point for the same radius, and because the radius can be either positive or negative, you can express the point with polar coordinates in many ways

https://www.whitman.edu/mathematics/calculus_online/section10.01.html

10.1 Polar Coordinates [Jump to exercises]

    

Collapse menu 1 Analytic Geometry 2 Instantaneous Rate of Change: The Derivative 3 Rules for Finding Derivatives 4 Transcendental Functions 5 Curve Sketching 6 Applications of the Derivative 7 Integration 8 Techniques of Integration 9 Applications of Integration 10 Polar Coordinates, Parametric Equations 1. Polar Coordinates 2. Slopes in polar coordinates 3. Areas in polar coordinates 4. Parametric Equations 5. Calculus with Parametric Equations 11 Sequences and Series 12 Three Dimensions 13 Vector Functions 14 Partial Differentiation 15 Multiple Integration 16 Vector Calculus 17 Differential Equations 18 Useful formulas

Coordinate systems are tools that let us use algebraic methods to understand geometry. While the rectangular (also called Cartesian) coordinates that we have been using are the most common, some problems are easier to analyze in alternate coordinate systems.

A coordinate system is a scheme that allows us to identify any point in the plane or in threedimensional space by a set of numbers. In rectangular coordinates these numbers are interpreted, roughly speaking, as the lengths of the sides of a rectangle. In polar coordinatesa point in the plane is identified by a pair of numbers (r,θ)(r,θ). The number θθ measures the angle between the positive xx-axis and a ray that goes through the point, as shown in figure 10.1.1; the number rr measures the distance from the origin to the point. Figure 10.1.1 shows the point with rectangular coordinates (1,3–√)(1,3) and polar coordinates (2,π/3)(2,π/3), 2 units from the origin and π/3π/3 radians from the positive xx-axis. 11 3–√3 (2,π/3)(2,π/3) Figure 10.1.1. Polar coordinates of the point (1,3–√)(1,3).

Just as we describe curves in the plane using equations involving xx and yy, so can we describe curves using equations involving rr and θθ. Most common are equations of the form r=f(θ)r=f(θ). Example 10.1.1 Graph the curve given by r=2r=2. All points with r=2r=2 are at distance 2 from the origin, so r=2r=2 describes the circle of radius 2 with center at the origin. Example 10.1.2 Graph the curve given by r=1+cosθr=1+cos⁡θ. We first consider y=1+cosxy=1+cos⁡x, as in figure 10.1.2. As θθ goes through the values in [0,2π][0,2π], the value of rr tracks the value of yy, forming the "cardioid'' shape of figure 10.1.2. For example, when θ=π/2θ=π/2, r=1+cos(π/2)=1r=1+cos⁡(π/2)=1, so we graph the point at distance 1 from the origin along the positive yy-axis, which is at an angle of π/2π/2 from the positive xx-axis. When θ=7π/4θ=7π/4, r=1+cos(7π/4)=1+2– √/2≈1.71r=1+cos⁡(7π/4)=1+2/2≈1.71, and the corresponding point appears in the fourth quadrant. This illustrates one of the potential benefits of using polar coordinates: the equation for this curve in rectangular coordinates would be quite complicated. π/2π/2 ππ 3π/23π/2 2π2π 11 22 Figure 10.1.2. A cardioid: y=1+cosxy=1+cos⁡x on the left, r=1+cosθr=1+cos⁡θ on the right.

Each point in the plane is associated with exactly one pair of numbers in the rectangular coordinate system; each point is associated with an infinite number of pairs in polar coordinates. In the cardioid example, we considered only the range 0≤θ≤2π0≤θ≤2π, and already there was a duplicate: (2,0)(2,0) and (2,2π)(2,2π) are the same point. Indeed, every value of θθ outside the interval [0,2π)[0,2π) duplicates a point on the curve r=1+cosθr=1+cos⁡θ when 0≤θ<2π0≤θ<2π. We can even make sense of polar coordinates like (−2,π/4)(−2,π/4): go to the direction π/4π/4 and then move a distance 2 in the opposite direction; see figure 10.1.3. As usual, a negative angle θθ means an angle measured clockwise from the positive xx-axis. The point in figure 10.1.3 also has coordinates (2,5π/4)(2,5π/4) and (2,−3π/4)(2,−3π/4). 11 11 −1−1 −1−1 22 22 −2−2 −2−2 π/4π/4 Figure 10.1.3. The point (−2,π/4)=(2,5π/4)=(2,−3π/4)(−2,π/4)=(2,5π/4)=(2,−3π/4) in polar coordinates.

The relationship between rectangular and polar coordinates is quite easy to understand. The point with polar coordinates (r,θ)(r,θ) has rectangular coordinates x=rcosθx=rcos⁡θ and y=rsinθy=rsin⁡θ; this follows immediately from the definition of the sine and cosine functions. Using figure 10.1.3 as an example, the point shown has rectangular coordinatesx=(−2)cos(π/4)=−2– √≈1.4142x=(−2)cos⁡(π/4)=−2≈1.4142 and y=(−2)sin(π/4)=−2–√y=(−2)sin⁡(π/4)=−2. This makes it very easy to convert equations from rectangular to polar coordinates. Example 10.1.3 Find the equation of the line y=3x+2y=3x+2 in polar coordinates. We merely substitute: rsinθ=3rcosθ+2rsin⁡θ=3rcos⁡θ+2, or r=2sinθ−3cosθr=2sin⁡θ−3cos⁡θ. Example 10.1.4 Find the equation of the circle (x−1/2)2+y2=1/4(x−1/2)2+y2=1/4 in polar coordinates. Again substituting: (rcosθ−1/2)2+r2sin2θ=1/4(rcos⁡θ−1/2)2+r2sin2⁡θ=1/4. A bit of algebra turns this into r=cos(t)r=cos⁡(t). You should try plotting a few (r,θ)(r,θ) values to convince yourself that this makes sense.

Example 10.1.5 Graph the polar equation r=θr=θ. Here the distance from the origin exactly matches the angle, so a bit of thought makes it clear that when θ≥0θ≥0 we get the spiral of Archimedes in figure 10.1.4. When θ<0θ<0, rr is also negative, and so the full graph is the right hand picture in the figure. (−1,−1)(−1,−1) (−π/2,−π/2)(−π/2,−π/2) (−π,−π)(−π,−π) (−2π,−2π)(−2π,−2π)

(1,1)(1,1) (π/2,π/2)(π/2,π/2) (π,π)(π,π) (2π,2π)(2π,2π)

Figure 10.1.4. The spiral of Archimedes and the full graph of r=θr=θ.

Converting polar equations to rectangular equations can be somewhat trickier, and graphing polar equations directly is also not always easy. Example 10.1.6 Graph r=2sinθr=2sin⁡θ. Because the sine is periodic, we know that we will get the entire curve for values of θθ in [0,2π)[0,2π). As θθ runs from 0 to π/2π/2, rr increases from 0 to 2. Then as θθ continues to ππ, rr decreases again to 0. When θθ runs from ππ to 2π2π, rr is negative, and it is not hard to see that the first part of the curve is simply traced out again, so in fact we get the whole curve for values of θθ in [0,π)[0,π). Thus, the curve looks something like figure 10.1.5. Now, this suggests that the curve could possibly be a circle, and if it is, it would have to be the circle x2+(y−1)2=1x2+(y−1)2=1. Having made this guess, we can easily check it. First we substitute for xx and yy to get (rcosθ)2+(rsinθ−1)2=1(rcos⁡θ)2+(rsin⁡θ−1)2=1; expanding and simplifying does indeed turn this into r=2sinθr=2sin⁡θ. π/2π/2 11 −1−1 ππ

11 11 −1−1

Figure 10.1.5. Graph of r=2sinθr=2sin⁡θ. You can drag the red point in the graph on the left, and the corresponding point on the right will follow.

Exercises 10.1 Ex 10.1.1 Plot these polar coordinate points on one graph: (2,π/3)(2,π/3), (−3,π/2)(−3,π/2), (−2,−π/4)(−2,−π/4), (1/2,π)(1/2,π), (1,4π/3)(1,4π/3), (0,3 π/2)(0,3π/2). Find an equation in polar coordinates that has the same graph as the given equation in rectangular coordinates.

Ex 10.1.2 y=3xy=3x (answer) Ex 10.1.3 y=−4y=−4 (answer) Ex 10.1.4 xy2=1xy2=1 (answer) Ex 10.1.5 x2+y2=5x2+y2=5 (answer) Ex 10.1.6 y=x3y=x3 (answer) Ex 10.1.7 y=sinxy=sin⁡x (answer) Ex 10.1.8 y=5x+2y=5x+2 (answer) Ex 10.1.9 x=2x=2 (answer) Ex 10.1.10 y=x2+1y=x2+1 (answer) Ex 10.1.11 y=3x2−2xy=3x2−2x (answer) Ex 10.1.12 y=x2+y2y=x2+y2 (answer) Sketch the curve. Ex 10.1.13 r=cosθr=cos⁡θ Ex 10.1.14 r=sin(θ+π/4)r=sin⁡(θ+π/4) Ex 10.1.15 r=−secθr=−sec⁡θ Ex 10.1.16 r=θ/2r=θ/2, θ≥0θ≥0 Ex 10.1.17 r=1+θ1/π2r=1+θ1/π2 Ex 10.1.18 r=cotθcscθr=cot⁡θcsc⁡θ Ex 10.1.19 r=1sinθ+cosθr=1sin⁡θ+cos⁡θ Ex 10.1.20 r2=−2secθcscθr2=−2sec⁡θcsc⁡θ

In the exercises below, find an equation in rectangular coordinates that has the same graph as the given equation in polar coordinates. Ex 10.1.21 r=sin(3θ)r=sin⁡(3θ) (answer)

(x2+y2)2=4x2y−(x2+y2)y(x2+y2)2=4x2y−(x2+y2)y Ex 10.1.22 r=sin2θr=sin2⁡θ (answer)

(x2+y2)3/2=y2(x2+y2)3/2=y2 Ex 10.1.23 r=secθcscθr=sec⁡θcsc⁡θ (answer)

x2+y2=x2y2x2+y2=x2y2 Ex 10.1.24 r=tanθr=tan⁡θ (answer)

x4+x2y2=y2x4+x2y2=y2

https://www.ck12.org/trigonometry/plot-polar-coordinates/lesson/Polar-Coordinates-MAT-ALY/ https://www.ck12.org/trigonometry/plot-polar-coordinates/lesson/Polar-Coordinates-MAT-ALY/

Everyone has dreamed of flying at one time or another. Not only would there be much less traffic to worry about, but directions would be so much simpler! Walking or driving: "Go East 2 blocks, turn left, then North 6 blocks. Wait for the train. Turn right, East 3 more blocks, careful of the cow! Turn left, go North 4 more blocks and park." Flying: "Fly 30 deg East of North for a little less than 11 and 1/4 blocks. Land." Nice daydream, but what does it have to do with polar coordinates?

Polar Coordinates The polar coordinate system is an alternative to the Cartesian coordinate system you have used in the past to graph functions. The polar coordinate system is specialized for visualizing and manipulating angles.

Angles are identified by travelling counter-clockwise around the circular graph from the 0 deg line, or r-axis (where the + x axis would be) to a specified angle.

To plot a specific point, first go along the r-axis by r units. Then, rotate counterclockwise by the given angle, commonly represented "θ". Be careful to use the correct units for the angle measure (either radians or degrees).

Radians Usually polar plots are done with radians (especially if they include trigonometric functions), but sometimes degrees are used. A radian is the angle formed between the r axis and a polar axis drawn to meet a section of the circumference that is the same length as the radius of a circle. Given that the circumference of a circle is 2π⋅r, and since r is the radius, that means there are 2π radians in a complete circle, and 1π radians in 1/2 of a circle. If 1/2 of a circle is π radians, and is 180 deg, that means that there are 180π degrees in each radian.

That translates to approximately 57.3 degrees = 1 radian.

Graphing Using Technology Polar equations can be graphed using a graphing calculator: With the graphing calculator- go to MODE. There select RADIAN for the angle measure and POL (for Polar) on the FUNC(function)line. When Y = is pressed, note that the equation has changed from y = to r = . There input the polar equation. After pressing graph, if you can’t see the full graph, adjust x- and y- max/min, etc in WINDOW.

Examples Example 1 Plot the points on a polar coordinate graph: Point A (2,π3), Point B (4,135o), and Point C (−2,π6).

Below is the pole, polar axis and the points A, B and C.

Example 2 Plot the following points.

a. b. c. d. e.

(4,30o) (2.5,π) (−1,π3) (3,5π6) (−2,300o)

Example 3 Use a graphing calculator or plotting program to plot the following equations. a.

r=1+3sinθ

b.

r=1+2cosθ

Review the steps above under graphing using technology if you are having trouble.

Example 4 Convert from radians to degrees. Recall that πrad=180o and 1rad=180π≈57.3o.

π2 If πrad=180o then π2rad=90o a.

5.17 If 1rad≈57.3o then 5.17rad≈296o b.

3π2 If πrad=180o then 3π2rad=270o c.

Example 5 Convert from degrees to radians. Recall that 180oπ=57.3o≈1rad.

251o If 57.3o≈1rad then 251o≈4.38rad≈1.4πrad a.

360o If 57.3o≈1rad then 360o≈6.28rad b.

327o If 57.3o≈1rad then 327o57.3o≈5.71rad c.

Example 6 Convert from degrees to radians, answer in terms of π. Recall that 2πrad=360o and therefore πrad=180o.

90o If πrad=180o then π2rad=90o a.

270o If πrad=180o and π2rad=90o then 112πrad→32π→3π2rad=270o b.

45o If π2rad=90o then π4rad=45o c.

Review 1. Why can a point on the plane not be labeled using a unique ordered pair (r,θ). 2. Explain how to graph (r,θ) if r<0 and/or θ>360.

Graph each point on the polar plane. 3. 4. 5. 6. 7.

A (6,145o) B (−2,13π6) C (74,−210o) D (5,π2) E (3.5,−π8)

Name two other pairs of polar coordinates for each point. 8. (1.5,170o) 9. (−5,π−3) 10. (3,305o)

Graph each polar equation. 11. r=3 12. θ=π5 13. r=15.5 14. r=1.5 15. θ=−175o

Find the distance between the given points. 16. P1(5,π2) and P2(7,3π9) 17. P1(1.3,−52o) and P2(−13.6,−162o) 18. P1(3,250o) and P2(7,90o)

The Polar Coordinate System https://courses.lumenlearning.com/boundlessalgebra/chapter/the-polar-coordinate-system/

Introduction to the Polar Coordinate System The polar coordinate system is an alternate coordinate system where the two variables are rr and θθ, instead of xx and yy.

LEARNING OBJECTIVES Discuss the characteristics of the polar coordinate system

KEY TAKEAWAYS Key Points 

A polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.

Key Terms    

radius: A distance measured from the pole. angular coordinate: An angle measured from the polar axis, usually counter-clockwise. pole: The reference point of the polar graph. polar axis: A ray from the pole in the reference direction.

Introduction of Polar Coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. When we think about plotting points in the plane, we usually think of rectangular coordinates (x,y)(x,y) in the Cartesian coordinate plane. However, there are other ways of writing a coordinate pair and other types of grid systems. Polar coordinates are points labeled (r,θ)(r,θ) and plotted on a polar grid. The polar grid is represented as a series of concentric circles radiating out from the pole, or the origin of the coordinate plane. The reference point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is called the angular coordinate, polar angle, or azimuth. The radial coordinate is often denoted by rr or ρρ, and the angular coordinate by ϕϕ, θθ, or tt.

Examples of Polar Coordinates: Points in the polar coordinate system with pole 00 and polar axis LL. In green, the point with radial coordinate 33 and angular coordinate 6060 degrees or (3,60∘)(3,60∘). In blue, the point (4,210∘)(4,210∘).

Polar Graph Paper: A polar grid with several angles labeled in degrees

Angles in polar notation are generally expressed in either degrees or radians ( 2π2π rad being equal to 360∘360∘). Degrees are traditionally used in navigation, surveying, and many applied disciplines, while radians are more common in mathematics and mathematical physics. In many contexts, a positive angular coordinate means that the angle ϕϕ is measured counterclockwise from the axis. In mathematical literature, the polar axis is often drawn horizontal and pointing to the right. Plotting Points Using Polar Coordinates The polar grid is scaled as the unit circle with the positive xx–axis now viewed as the polar axis and the origin as the pole. The first coordinate rr is the radius or length of the

directed line segment from the pole. The angle θθ, measured in radians, indicates the direction of rr. We move counterclockwise from the polar axis by an angle of θθ,and measure a directed line segment the length of rr in the direction of θθ. Even though we measure θθ first and then rr, the polar point is written with the rr-coordinate first. For example, to plot the point (2,π4)(2,π4),we would move π4π4 units in the counterclockwise direction and then a length of 22 from the pole. This point is plotted on the grid in Figure.

Plotting a point on a Polar Grid: Plot of the point (2,π4)(2,π4),by moving π4π4 units in the counterclockwise direction and then a length of 22 from the pole.

Uniqueness of polar coordinates Adding any number of full turns (360∘360∘ or 2π2π radians) to the angular coordinate does not change the corresponding direction. Also, a negative radial coordinate is best interpreted as the corresponding positive distance measured in the opposite direction. Therefore, the same point can be expressed with an infinite number of different polar coordinates(r,ϕ±n⋅360°r,ϕ±n⋅360°) or (−r,ϕ±(2n+1)⋅180°−r,ϕ±(2n+1)⋅180°), where nn is any integer. Moreover, the pole itself can be expressed as (0,ϕ0,ϕ) for any angle ϕϕ.

Converting Between Polar and Cartesian Coordinates

Polar and Cartesian coordinates can be interconverted using the Pythagorean Theorem and trigonometry.

LEARNING OBJECTIVES Derive and use the formulae for converting between Polar and Cartesian coordinates

KEY TAKEAWAYS Key Points 

To convert from polar to rectangular (Cartesian) coordinates use the following formulas (derived from their trigonometric function definitions):



cosθ=xr→x=rcosθsinθ=yr→y=rsinθr2=x2+y2tanθ=yxcos⁡θ=xr→x=rcos⁡θsin⁡θ=y r→y=rsin⁡θr2=x2+y2tan⁡θ=yx

Polar Coordinates to Rectangular (Cartesian) Coordinates When given a set of polar coordinates, we may need to convert them to rectangular coordinates. To do so, we can recall the relationships that exist among the variables  xx, yy, rr, and θθ, from the definitions of cosθcos⁡θ and sinθsin⁡θ. Solving for the variables xx and yy yields the following formulas:

cosθ=xr⇒x=rcosθcos⁡θ=xr⇒x=rcos⁡θ sinθ=yr⇒y=rsinθsin⁡θ=yr⇒y=rsin⁡θ An easy way to remember the equations above is to think of cosθcos⁡θ as the adjacent side over the hypotenuse and sinθsin⁡θ as the opposite side over the hypotenuse. Dropping a perpendicular from the point in the plane to the xx–axis forms a right triangle, as illustrated in Figure below.

Trigonometry Right Triangle: A right triangle with rectangular (Cartesian) coordinates and equivalent polar coordinates.

To convert polar coordinates (r,θ)(r,θ) to rectangular coordinates (x,y)(x,y) follow these steps: 1) Write cosθ=xr⇒x=rcosθcos⁡θ=xr⇒x=rcos⁡θ and sinθ=yr⇒y=rsinθsin⁡θ=yr⇒y=rsin⁡ θ. 2) Evaluate cosθcos⁡θ and sinθsin⁡θ. 3) Multiply cosθcos⁡θ by rr to find the xx-coordinate of the rectangular form. 4) Multiply sinθsin⁡θ by rr to find the yy-coordinate of the rectangular form. Example: Write the polar coordinates (3,π2)(3,π2) as rectangular coordinates.

x=rcosθ=3cosπ2=0x=rcos⁡θ=3cosπ2=0 y=rsinθ=3sinπ2=3y=rsin⁡θ=3sin⁡π2=3 The rectangular coordinates are (0,3)(0,3).

Polar and Coordinate Grid of Equivalent Points: The rectangular coordinate (0,3)(0,3) is the same as the polar coordinate (3,π2)(3,π2) as plotted on the two grids above.

Rectangular (Cartesian) Coordinates to Polar Coordinates To convert rectangular coordinates to polar coordinates, we will use two other familiar relationships. With this conversion, however, we need to be aware that a set of rectangular coordinates will yield more than one polar point. Converting from rectangular coordinates to polar coordinates requires the use of one or more of the relationships illustrated below. Recall:

cosθ=xr⇒x=rcosθsinθ=yr⇒y=rsinθr2=x2+y2tanθ=yxcos⁡θ=xr⇒x=rcos⁡θsin⁡θ=yr⇒y =rsin⁡θr2=x2+y2tan⁡θ=yx

Trigonometry Right Triangle: A right triangle with rectangular (Cartesian) coordinates and equivalent polar coordinates.

Example: Convert the rectangular coordinates (3,3)(3,3) to polar coordinates. We are given the values of xx and yy and need to solve for θθ and rr. Start by solving for θθ using the tantan function:

tanθ=yx=33=1tan⁡θ=yx=33=1 So:

θ=tan−1(1)=π4θ=tan−1⁡(1)=π4 Next substitute the values of xx and yy into the formula r2=x2+y2r2=x2+y2 and solve for rr.

r2=x2+y2=32+32=18r2=x2+y2=32+32=18 So:

r=√18=3√2r=18=32 The polar coordinates are (3√2,π4)(32,π4). Note that r2=18r2=18 implies r=±√18r=±18. We chose to ignore the negative rr value. Also note that tan−1(1)tan−1⁡(1) has many answers. This corresponds to the nonuniqueness of polar coordinates. Multiple sets of polar coordinates can have the same

location as our first solution. For example, the points

(−3√2,5π4)(−32,5π4) and (3√2,−7π2)(32,−7π2) will coincide with the original solution of (3√2,π4)(32,π4).

Conics in Polar Coordinates Polar coordinates allow conic sections to be expressed in an elegant way.

LEARNING OBJECTIVES Describe the equations for different conic sections in polar coordinates

KEY TAKEAWAYS Key Points 

Conic sections have several key features which define their polar equation; foci, eccentricity, and a directrix.



All conic sections have the same basic equation in polar coordinates, which demonstrates a connection between all of them.

Key Terms  

eccentricity: A measure of deviation from a prescribed curve. directrix: A fixed line used to described a curve.

Defining a Conic Previously, we learned how a parabola is defined by the focus (a fixed point ) and the directrix (a fixed line).

Parts of a Parabola: Consider the parabola x=2+y2x=2+y2. Any conic may be determined by three characteristics: a single focus, a fixed line called the directrix, and the ratio of the distances of each to a point on the graph.

We can define any conic in the polar coordinate system in terms of a fixed point, the focus P(r,θ)P(r,θ) at the pole, and a line, the directrix, which is perpendicular to the polar axis. For a conic with eccentricity ee, 1. If 0≤e<10≤e<1, the conic is an ellipse. 2. If e=1e=1, the conic is a parabola. 3. If e>1e>1, the conic is an hyperbola .

With this definition, we may now define a conic in terms of the directrix:  x=±px=±p, the eccentricity ee, and the angle θθ. Thus, each conic may be written as a polar equation in terms of rr and θθ. For a conic with a focus at the origin, if the directrix is x=±px=±p, where pp is a positive real number, and the eccentricity is a positive real number ee, the conic has a polar equation:

r=e⋅p1±e⋅cosθr=e⋅p1±e⋅cos⁡θ For a conic with a focus at the origin, if the directrix is y=±py=±p, where pp is a positive real number, and the eccentricity is a positive real number ee, the conic has a polar equation:

r=e⋅p1±e⋅sinθr=e⋅p1±e⋅sin⁡θ

Other Curves in Polar Coordinates Some curves have a simple expression in polar coordinates, whereas they would be very complex to represent in Cartesian coordinates.

LEARNING OBJECTIVES Describe the equations for spirals and roses in polar coordinates

KEY TAKEAWAYS Key Points 



The formulas that generate the graph of a rose curve are given by: r=acosnθr=acos⁡nθ  and r=asinnθr=asin⁡nθ  where a≠0a≠0. If  nn is even, the curve has 2n2n petals. If  nn is odd, the curve has nn petals. The formula that generates the graph of the Archimedes’ spiral is given by: r=θr=θ for  θ≥0θ≥0. As  θθ increases,  rr increases at a constant rate in an everwidening, never-ending, spiraling path.

Key Terms  

Archimedes’ spiral: A curve given by an equation of the form r=a+bθr=a+bθ rose curve: A curve given by an equation of the form r=acosnθr=acos⁡nθ or r=asinnθr=asin⁡nθ

To graph in the rectangular coordinate system we construct a table of xx and yy  values. To graph in the polar coordinate system we construct a table of rr and θθ values. We enter values of θθ into a polar equation and calculate rr. However, using the properties of symmetry and finding key values of θθ and rr means fewer calculations will be needed. Investigating Rose Curves Polar equations can be used to generate unique graphs. The following type of polar equation produces a petal-like shape called a rose curve. Although the graphs look complex, a simple polar equation generates the pattern. The formulas that generate the graph of a rose curve are given by:

r=a⋅cos(nθ)andr=a⋅sin(nθ)wherea≠0r=a⋅cos⁡(nθ)andr=a⋅sin⁡(nθ)wherea≠0

If nn is even, the curve has 2n2n petals. If nn is odd, the curve has nn petals.

Rose Curves: Complex graphs generated by the simple polar curves:r=acosnθr=acos⁡nθ and r=asinnθr=asin⁡nθ where a≠0a≠0.

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Investigating the Archimedes’ Spiral Archimedes’ spiral is named for its discoverer, the Greek mathematician Archimedes (c.287BCE−c.212BCEc.287BCE−c.212BCE), who is credited with numerous discoveries in the fields of geometry and mechanics. The formula that generates the graph of the Archimedes’ spiral is given by:

r=a+bθforθ≥0r=a+bθforθ≥0 As θθ increases, rr increases at a constant rate in an ever-widening, never-ending, spiraling path.

rose

Archimedes’ Spiral: The formula that generates the graph of a spiral is r=θr=θ for θ≥0θ≥0.

https://betterlesson.com/lesson/595128/the-polar-coordinate-system

Objective SWBAT graph polar coordinates and convert points and equations between polar and rectangular form.

Big Idea How is the rectangular and polar coordinate systems related?

Bell work 5 MINUTES

Over the next few lessons I will be working with an idea is not specifically discussed in the Common Core standards. By teaching the polar coordinate system and its equations I am able to apply many of the previous topics we learned in new and insightful ways. For example, we will review trigonometric concepts, such as trigonometric identities and real valued functions with points on the coordinate plane, when learning the polar coordinate system.

I begin today's lesson by reviewing how to graph points when of real-valued functions. Polar graphing has a very similar structure, the only aspect that is different is that the ordered pair is (r,theta) instead of (x,y) To begin, I have one student in the class give an example by graphing the point. I will then review with the students any idea they may have forgotten since the real-valued unit last semester.

How to graph polar coordinates? 10 MINUTES

I now share the definition provided in the Larson textbook "Precalculus with Limits, 2nd ed." We analyze together the definition that written in the book. My goal is for students to see that you can relate a rectangular coordinate plane with a polar coordinate system. I give every student a copy of a polar coordinate (These are shared at the end of this section.) Some questions I ask students about the polar graph are:  

If you were to label an x and y axis on this graph, where would they be? The pole is the center point on this graph, how would you label this point as an (x,y) point? What would it be for (r,theta)? After examining a plane graph, I have the students practice by graphing some points. I spend time on points that have negative r values since this is usually one of the most challenging parts of a polar coordinate system. I review the concept of how the negative value of an x or y coordinate reveals the direction the point is from the origin. A negative value on the r coordinate also gives the direction of the point. Once the students see how to graph a point, I put a point in polar form on the board and ask the students to find another way to label the point. I let the students consider this question and discuss what a possible solution. I put the students' ideas on the board and ask each student to explain how they arrived at their answer. If a student does not use a -r, then I will ask what would the label be if the r is negative.

Lastly, I ask "How many ways can we label a specific point in polar coordinates?" Inevitably, I want students to realize that there are infinite many ways. It is important to reinforce that the value is not just the angle to 2pi. http://www.mathguide.com/lessons3/PolarC.html

Polar Coordinates Home > Lessons > Polar Coordinates

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Introduction This lesson page will inform you about polar coordinates. Here are the sections within this lesson page:      

Introduction Graphing Polar Coordinates Converting from Polar to Rectangular Coordinates Converting from Rectangular to Polar Coordinates Instructional Videos Related Lessons

Introduction When graphing a point in the Cartesian coordinate system, we do so using horizontal and vertical movements. This is similar to moving through a city that has nothing but East-West and North-South streets. There is another system for graphing points in a plane, called the polar system. This system has its points defined by their distance from the origin and their angle from the positive x-axis. The following section will explain how to graph polar points.

Graphing Polar Coordinates Here are two examples of graphing polar coordinates. Example 1: Graph (3, 45°). To graph this point, imagine starting at the origin and looking down the positive x-axis. It is as if you are standing at the origin and looking east. Next, rotate 45° counter-clockwise and then move 3 units from the origin.

Example 2: Graph (2, 210°). Starting at the origin while facing east, turn 210° counter-clockwise. Then, move 2 units from the origin. This is what the result looks like.

ideo: Polar Coordinates and Rectangular Conversions

Converting from Polar to Rectangular Coordinates To convert from rectangular to polar coordinates, we have to use two equations. Here are those equations.

The equations relate to the definition of the cosine and sine functions.

Cosine is defined as adjacent (x) over hypotenuse (r). Likewise, sine is defined as opposite (y) over hypotenuse (r). With a little algebra, we can solve for ‘x’ and for ‘y’ to achieve the conversion equations shown above. Here are two examples. Example 1: Convert (2,135°) to rectangular coordinates. For a purely algebraic solution, we can solve for the x-value and the y-value as follows, keeping in mind that r=2 and θ= 135°.

So, the point in rectangular form is…

Example 2: Convert (-1,330°) to rectangular coordinates. Solve for the x-value and the y-value using the equations. Remember, r=-1 and θ= 330°.

So, the point in rectangular form is…

ideo: Polar Coordinates and Rectangular Conversions

Converting from Rectangular to Polar Coordinates To convert from rectangular to polar coordinates requires different equations. Here are those equations.

To understand the genesis of these equations, examine this diagram.

The relationship between the x, y, and r-variables should be familiar. Since this is a right triangle, we can employ The Pythagorean Theorem, which is the first of the two conversion equations. Using knowledge of trigonometry, we can see the tangent of theta is equal to the opposite (y) over adjacent (x) sides, which is the second conversion equation. Now, let us look at two examples to see how these conversions are done. Example 1: Convert (5,-3) to polar form, rounded to the nearest tenth. It is helpful to get a diagram to see what is going on. Here is the graph of the rectangular point.

To get the distance the point is from the origin, which is the r-value, we will use the first conversion equation, like so.

Remember, this angle is the reference angle. Since the angle exists in the fourth quadrant, we have to account for the traditional trigonometric angle relative to the positive x-axis with a counter-clockwise motion. Therefore, the angle is 360° - 31° = 329.0°. So, the final answer, written as (r, θ), is…

Example 2: Convert (-1, √3) from rectangular form to polar form. Here is a diagram of the point in the second quadrant.

It is unnecessary to calculate the length of the hypotenuse if you recognize this special right triangle. Assuming you do not recognize the triangle, let us view the calculation using the first conversion equation.

Now, we must calculate the angle using the second conversion equation (if you do not recognize the special right triangle). Since we already know the angle exists in the second quadrant, only positive values are being used.

Remember, this is the reference angle not the angle to the positive x-axis. The angle to the positive x-axis (rotating in the typical counter-clockwise fashion) is 120°. Last, we need to write the point in (r, θ) form as…

ideo: Polar Coordinates and Rectangular Conversions

Instructional Videos This instructional video will help you understand polar coordinates. ideo: Polar Coordinates and Rectangular Conversions

Related Lessons Review these related lessons. esson: Trigonometry with Right Triangles esson: Trigonometric Angles esson: Vectors

http://www.shelovesmath.com/trigonometry/polar-graphs/ So far, we’ve plotted points using rectangular (or Cartesian) coordinates, since the points since we are going back and forth xunits, and up and down y units.

Plotting Points Using Polar Coordinates

In the Polar Coordinate System, we go around the origin or the pole a certain distance out, and a certain angle from the positivex-axis:

The ordered pairs, called polar coordinates, are in the form (r,θ), with r being the number of units from the origin or pole (if r > 0), like a radius of a circle, and θ being the angle (in degrees or radians) formed by the ray on the positive x-axis (polar axis), going counter-clockwise. If r < 0, the point is r units (like a radius) in the opposite direction (across the origin or pole) of the angle θ. If θ<0, you go clockwise with the angle, starting with the positive x-axis. So to plot the point, you typically circle around the positive x-axis θ degrees first, and then go out from the origin or pole r units (if r is negative, go the other way (180°) r units). Here a polar graph with some points on it. Note that we typically count in increments of 15°, or π12.

For a point (r,θ), do you see how you always go counter-clockwise (or clockwise, if you have a negative angle) until you reach the angle you want, and then out from the center r units, if r is positive? If r is negative, you go in the opposite direction from the angle r units. If both r and the angle θ are negative, you have to make sure you go clockwise to get the angle, but in the opposite direction r units. You may be asked to rename a point in several different ways, for example, between [−2π,2π) or [−360∘,360∘). For example, if we wanted to rename the point (6,240∘) three other different ways between [−360∘,360∘), by looking at the graph above, we’d get (−6,60∘)(make r negative and subtract 180°), (6,−120∘) (subtract 360°), and (−6,−240∘) (make both negative). (Remember that 240 and –120, and 60 and –240 are co-terminal angles). To get these, if the first number (r) is negative, you want to go in the opposite direction, and if the angle is negative, you want to go clockwise instead of counterclockwise from the positive x-axis.