<< Chapter < Page Chapter >> Page >
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!
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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!
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Write the polar coordinates ( 1 , 2 π 3 ) as rectangular coordinates.

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

Got questions? Get instant answers now!

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!
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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 golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 3

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask