Tutoring Precalculus – Polar Coordinates Intro 9 Examples


Tutoring Precalculus Polar Coordinates Intro 9 Examples.


What are polar Coordinates and

Why do we need them?

What are polar coordinates?

When you think of polar coordinates, something about the North Pole, polar bears and the arctic come to mind. They are not the coordinates dedicated to finding polar bears!  However the idea for polar coordinates can be thought of as the coordinates “surrounding” the pole or the origin.

What do we need them for?

Some of the real-life uses of polar coordinates include avoiding collisions between vessels and other ships or natural obstructions, guiding industrial robots in various production applications and calculating groundwater flow in radially symmetric wells. Polar coordinates can also be used to determine the best audio pickup patterns for cardioid microphones. Calculations involving aircraft navigation, gravitational fields and radio antennae are additional applications in which polar coordinates are used.


The first number u is taken to be a distance from the origin and the second number v is taken to be an angle (usually in radians). To be explicit about this, we will denote the pair as (r, θ) instead of (u, v). The numbers r and θ can be positive, negative or zero.



The pole (or origin)
and is labeled O.

Plotting Polar Coordinates.


a)  (-2, π/2)  b)  (3, –)     c)  (-1, )


Find the location of the angle: if the angle is positive use counterclockwise orientation, if the angle is negative use clockwise.

π/2 is 90 degrees.  90 degrees is the top half of the 180 x axis. To find r=-2 we travel the opposite direction from the origin and end up in the lower half of the coordinate system.




Find the location of the angle: the angle is negative which mean we need to use clockwise orientation.

Lets divide pi=180=line into six equal parts and locate –pi/6= -30 degrees in the clockwise fashion(going underneath the x axis.

If the radius r was negative, r =-3 we would travel in the opposite direction after we locate the angle. But since the radius r=3 we do not travel in the opposite direction



Let’s start with the angle, is it positive or negative? It is negative , which means we go underneath the x axis two divisions. pi is divided by 3 times in this case. R=-1 is negative which means we travel in the opposite direction to find the point in Quadrant I.



  • There is a possibility of a two or more distinct points (polar coordinates) to be on the same spot.

(r1,θ) , (r2, θ) If r1=r2.


(2, pi/6) is on the same spot as (2, 13pi/6).

The reason this is true is that adding a full circle rotation to the angle will result in the same spot.

Try to put your finger on the number 2 and circulate your finger until it reaches 2 again (this is a full circle). Now add 30 degrees to that and you will end up on the same spot.

2) For negative angles add a pi=180 degree, if r is the opposite sign.

  • Coordinate (0, θ) all map to zero.

Converting from Rectangular Coordinates to polar Coordinates.


Dropping the perpendicular line to an x axis in any quadrant will always create a 90 degree angle.  Using the Pythagorean theorem r is the hypotenuse, Y is the side opposite the  , X is the adjacent side to the  .



We use the convention that an angle is:


  • Positive—if measured in the counterclockwise
    direction from the polar axis.
  • Negative—if measured inthe clockwise
    direction from the polar axis.
  • We agree that, as shown, the points (–r,θ)
    and (r,θ) lie on the same line through O
    and at the samedistance | r | fromO,but
    on oppositesides of O.



Convert the given rectangular coordinates into polar coordinates.

  1. (0, 10)     b. (6, -8)     c.(-4, 2)



Things to remember: Theta -the angle is always from the x axis.

Use the right triangle formulas, Pythagorean Theorem and trig function tan(theta). Since the x and y will be known always.

Locate the Point first on the coordinate axis. Plot the point in question.

Polar coordinates are expressed as a radius and an angle if you know both that means you solved the question.





Here the angle can be expressed as a positive angle by subtracting it from 360 degrees.

  1. Tan (8/6) = theta.Use the tangent formula to find the angle because opposite side 8 is known and adjacent side 6 is known tangent (theta) = opposite /adjacent .

The answer theta is positive but notice that the angle is underneath the x axis which means it is negative in respect to the x axis.




Converting from Polar Coordinates to Rectangular Coordinates.

To convert from polar to rectangular simply use the formulas for x and y.


  1. (20, 150)

Using a calculator:  This is the quickest and simplest method of getting the answer, although it will not yield the exact answer in cases with square roots.  For this example especially since it contains square root of 3.

Using parenthesis around the values is essential. MODE must be set to radians when using pi, degrees if using numbers. In this example we are using pi and the MODE is set to radians. To set the Mode on your calculator:

MODE make sure the word RADIAN is highlighted black.  Use the up and down arrow to reach the third line. If you want to select RADIAN move curser to the word RADIAN then hit enter, if you want to select the word DEGREE move move over to the word degree and hit enter. 2ndQUIT to escape.



If you are using 180 instead of pi which you can do, make sure you set your calculator to degree mode first. Notice how the answer in the second picture produces the right result only after we change the radian mode to degree mode.

Online Ti-84 Plus can be found here:



Using mental Math:

An exact answer can be found for this question, since the angle is a multiple of 30 degrees.

(See the chart below).


Sine function is negative in the quadrants where the y values are negative (Quadrants III, Quadrant IV). Cosine function is positive in Quadrants I,IV

Cosine of 150 is negative because the angle 150 is in the second quadrant where the cosine is negative. Using the value of 30 as a reference angle (see lesson on reference angles), the value of 30 is the same as the value of 150 except the sign is negative.

b. (75, -π)











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