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cos θ = x r x = r cos θ sin θ = y r y = r sin θ

Dropping a perpendicular from the point in the plane to the x- axis forms a right triangle, as illustrated in [link] . An easy way to remember the equations above is to think of cos θ as the adjacent side over the hypotenuse and sin θ as the opposite side over the hypotenuse.

Comparison between polar coordinates and rectangular coordinates. There is a right triangle plotted on the x,y axis. The sides are a horizontal line on the x-axis of length x, a vertical line extending from thex-axis to some point in quadrant 1, and a hypotenuse r extending from the origin to that same point in quadrant 1. The vertices are at the origin (0,0), some point along the x-axis at (x,0), and that point in quadrant 1. This last point is (x,y) or (r, theta), depending which system of coordinates you use.

Converting from polar coordinates to rectangular coordinates

To convert polar coordinates ( r , θ ) to rectangular coordinates ( x , y ) , let

cos θ = x r x = r cos θ
sin θ = y r y = r sin θ

Given polar coordinates, convert to rectangular coordinates.

  1. Given the polar coordinate ( r , θ ) , write x = r cos θ and y = r sin θ .
  2. Evaluate cos θ and sin θ .
  3. Multiply cos θ by r to find the x- coordinate of the rectangular form.
  4. Multiply sin θ by r to find the y- coordinate of the rectangular form.

Writing polar coordinates as rectangular coordinates

Write the polar coordinates ( 3 , π 2 ) as rectangular coordinates.

Use the equivalent relationships.

x = r cos θ x = 3 cos π 2 = 0 y = r sin θ y = 3 sin π 2 = 3

The rectangular coordinates are ( 0 , 3 ) . See [link] .

Illustration of (3, pi/2) in polar coordinates and (0,3) in rectangular coordinates - they are the same point!
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Writing polar coordinates as rectangular coordinates

Write the polar coordinates ( 2 , 0 ) as rectangular coordinates.

See [link] . Writing the polar coordinates as rectangular, we have

x = r cos θ x = −2 cos ( 0 ) = −2 y = r sin θ y = −2 sin ( 0 ) = 0

The rectangular coordinates are also ( 2 , 0 ) .

Illustration of (-2, 0) in polar coordinates and (-2,0) in rectangular coordinates - they are the same point!
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Write the polar coordinates ( 1 , 2 π 3 ) as rectangular coordinates.

( x , y ) = ( 1 2 , 3 2 )

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Converting from rectangular 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

Converting from rectangular coordinates to polar coordinates requires the use of one or more of the relationships illustrated in [link] .

cos θ = x r  or x = r cos θ sin θ = y r  or y = r sin θ r 2 = x 2 + y 2 tan θ = y x

Writing rectangular coordinates as polar coordinates

Convert the rectangular coordinates ( 3 , 3 ) to polar coordinates.

We see that the original point ( 3 , 3 ) is in the first quadrant. To find θ , use the formula tan θ = y x . This gives

tan θ = 3 3 tan θ = 1 tan −1 ( 1 ) = π 4

To find r , we substitute the values for x and y into the formula r = x 2 + y 2 . We know that r must be positive, as π 4 is in the first quadrant. Thus

r = 3 2 + 3 2 r = 9 + 9 r = 18 = 3 2

So, r = 3 2 and θ = π 4 , giving us the polar point ( 3 2 , π 4 ) . See [link] .

Illustration of (3rad2, pi/4) in polar coordinates and (3,3) in rectangular coordinates - they are the same point!
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Transforming equations between polar and rectangular forms

We can now convert coordinates between polar and rectangular form. Converting equations can be more difficult, but it can be beneficial to be able to convert between the two forms. Since there are a number of polar equations that cannot be expressed clearly in Cartesian form, and vice versa, we can use the same procedures we used to convert points between the coordinate systems. We can then use a graphing calculator to graph either the rectangular form or the polar form of the equation.

Given an equation in polar form, graph it using a graphing calculator.

  1. Change the MODE to POL , representing polar form.
  2. Press the Y= button to bring up a screen allowing the input of six equations: r 1 , r 2 , . . . , r 6 .
  3. Enter the polar equation, set equal to r .
  4. Press GRAPH .

Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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