# Is there an algorithms to determine correct angles

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1. Jun 4, 2017

### Vital

1. The problem statement, all variables and given/known data
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:
$r = cos(5\theta)$

2. Relevant 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.

3. The attempt at a solution
Thank you very much!

2. Jun 4, 2017

### 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 .

3. Jun 4, 2017

### 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)

Last edited: Jun 4, 2017
4. Jun 4, 2017

### Vital

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. :-)

5. Jun 4, 2017

### Vital

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.

6. Jun 4, 2017

### FactChecker

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.