Is there an algorithms to determine correct angles

[Vital]

Homework Statement

Hello!
I am struggling with plotting polar graphs manually (without any help of the calculator). My main unresolved issue is with finding correct values of theta in a given range.
For example, I have an equation:
$r = cos(5\theta)$

Homework Equations

and I know that the range I have to work with is [-π/5 , π/5]. Here are the values I get when arbitrary choose values of theta to compute r, then x and y.

But the problem is that I don't know of any algorithm that I could apply in any such case to choose correct values of theta - here I started with -π/5, took π/6, next π/10, and 0 for the first half.
I suppose there should be some algorithm or rules applicable for such choices.
Please, help me to fill that gap in my knowledge.

The Attempt at a Solution

Thank you very much!

 

[Nidum]

Increment the angle by fixed amounts starting at the lowest range value and ending at the highest range value . You have to decide what increment is appropriate for your application .

For example we could decide on 0.1 of π/5 as the increment .

Then starting from the lowest range value and incrementing the angle sequentially by 0.1 of π/5 we get :

- π/5 , - 0.9 x π/5 , - 0.8 x π/5 , .... , 0 , 0.1 x π/5 , 0.2 x π/5 , ... , π/5

Note though that it may be easier to get or make some proper polar graph paper and plot results directly rather than convert to Cartesian coordinates .

 

[FactChecker]

If you want to do quick plots from memory, you should use the angles inside sin() and cos() whose values you can remember. You should memorize the sin() and cos() of 0°, 30°, 45°, 60°, and 90°. Pick theta so that you are taking sin() and cos() of those angles.

(The sin() of 0°, 30°, 45°, 60°, and 90° is √0/2=0, √1/2=1/2, √2/2, √3/2, √4/2=1. And use √2/2 ≅ 1.414/2 = 0.707 and √3/2 ≅ 1.732/2 = 0.866)

 

[Vital]

Increment the angle by fixed amounts starting at the lowest range value and ending at the highest range value . You have to decide what increment is appropriate for your application .

For example we could decide on 0.1 of π/5 as the increment .

Then starting from the lowest range value and incrementing the angle sequentially by 0.1 of π/5 we get :

- π/5 , - 0.9 x π/5 , - 0.8 x π/5 , .... , 0 , 0.1 x π/5 , 0.2 x π/5 , ... , π/5

Note though that it may be easier to get or make some proper polar graph paper and plot results directly rather than convert to Cartesian coordinates .

Thank you very much. I will try to follow your advice, and see where it brings me to. :) As to direct plotting, well, no, I need to do the fundamental cycle first on r theta graph, and then convert to Cartesian; all manually, without any machines. :-)

 

[Vital]

If you want to do quick plots from memory, you should use the angles inside sin() and cos() whose values you can remember. You should memorize the sin() and cos() of 0°, 30°, 45°, 60°, and 90°. Pick theta so that you are taking sin() and cos() of those angles.

(The sin() of 0°, 30°, 45°, 60°, and 90° is √0/2=0, √1/2=1/2, √2/2, √3/2, √4/2=1. And use √2/2 ≅ 1.414/2 = 0.707 and √3/2 ≅ 1.732/2 = 0.866)

Yes, I know all these values pretty well - seems the only topic I definitely don't have issues with. I did try using these "standard" angles, but this is wrong in this case because the range of the given equation is 2π/5. So, I wondered what algorithm should be used that works in all cases.

 

[FactChecker]

Yes, I know all these values pretty well - seems the only topic I definitely don't have issues with. I did try using these "standard" angles, but this is wrong in this case because the range of the given equation is 2π/5. So, I wondered what algorithm should be used that works in all cases.

The equation is cos(5θ), so for instance √2/2 = cos(5θ) when 5θ = π/4 or θ = π/20. π/20 is well within the range ±π/4. So you can get a lot of points to plot within the range of ±π/4.