Compute unit circle coordinates (cos,sin) from given degree/radian?

[chromechris]

Hi, is there a way to mathematically come up with coordinates in a unit circle from a given radian/degree reference/coterminal angle? For example, if I am given a 30 degree angle I can look up a unit circle diagram and see that it's corresponding unit circle coordinates are (√3/2, 1/2), but is there another way of getting these coordinates through pure arithmetic using the given radian/degree angle?

 

[tkhunny]

Short Answer: No.
Longer Answer: You can't define "pure arithmetic".
Still Longer: Sure, you can plod through graphical methods and find as close an approximation as you would like, but is that "pure arithmetic"?

 

[chromechris]

What methods are you referring to?

 

[Dr.Peterson]

If you're asking how to compute sine and cosine of an angle in radians without using a calculator's trig buttons, the answer is that you could do what people did before computers to make trig tables. That was a lot of arithmetic!

Or, you could do what calculators do internally, which is to start with a small table of data and do some manipulations to get functions of any angle. That, too, is arithmetic.

Or you could use Taylor series or similar iterative methods. (Actually, all these methods are iterative.)

But those methods, by hand or by machine, only make approximations; they couldn't tell you that the cosine of pi/6 radians is exactly √3/2. That is done by geometrical reasoning.

 

[chromechris]

Got ya, thank you very much! I guess it's just better to work with geometrical reasoning by using square root amounts instead of decimal approximations.

 

[Dr.Peterson]

Got ya, thank you very much! I guess it's just better to work with geometrical reasoning by using square root amounts instead of decimal approximations.

... if what you want is an exact answer, and that's possible. Most sines and cosines can't be found exactly.

In real life, the decimals are far more appropriate. (And the calculator is far easier to use than the decades it takes to make your own tables.)

 

[Cubist]

I just found a very interesting Wikipedia page about a 7^th century method by Bhaskara I (click)

let p = x*(180-x), where x is in degrees, then sin(x) ≈ 4p / (40500 - p)

It stays remarkably accurate over 0 < x< 180, error less than 0.0017