Finding Points on the Unit Circle

[JungleJesus]

I am currently working on an implementation of a Symbolic Algebra system similar to existing products. In this system, I would like to be able to display the exact symbolic values of trigonometric functions for any given angle in radians.

ex: sin(PI/6) = "1/2"

My problem stems from obtaining the ratios from points on the unit circle, x**2 + y**2 = 1

I need to be able to calculate the point P(x,y) from the angle (a) on the unit circle without using any trigonometric functions because I will use those points to define the functions.

Some of the people I have asked suggested using calculus to find the coordinates. However, I don't know much about calculus or advanced trig.

Would Pythagorean's theory help in any way? Would it be more beneficial to use Calculus?

Anything to point me in the right direction would help greatly.

 

[Filip Larsen]

Maybe you can find something useful on http://en.wikipedia.org/wiki/List_of_trigonometric_identities

 

[hotvette]

Do you really want to express the trig function of any arbitrary angle (e.g. 1.23785763 radians) as a symbolic answer (e.g. root(xxx)/yyy)?

Also, are you saying you want to find the x,y coordinates of the point on the circle corresponding to the above arbitrary angle, without using any trig functions like sin or cos?

If so, seems the first is impossible unless you are dealing with only fixed increments of angle. Also, I think trigonomentric functions and circle geometry are inextricably connected.

Maybe you need to provide more explanation as to what you are trying to do.

 

[superflex]

You can do away with the whole unit circle business. Use Taylor series expansions for sin(A) and cos(A). It's an infinite series polynomial expansion, so you can choose how many terms you want to calculate and get accurate output to an arbitrary level of precision.

See the following:
http://en.wikipedia.org/wiki/Trigonometric_functions#Series_definitions"

 

[JungleJesus]

I guess I wasn't entirely clear. I don't want to find an approximation; I want the functions to output the Symbolic values of the functions. Trust me on this one, I'm looking for the kind of anwer you'd write on a trig test

ex: sin(PI/4) = "$$\sqrt{2}$$/2"

In order to accomplish this, I've been trying to use x² + y² = 1, but have not been able to determine the point (x,y) based on the inputted angle.

I thought of using the derivative with respect to x and solving for the angle. In doing so, I could manipulate the algebra to produce the x and y values which define the trig functions. However, I don't know how to connect the angle of the radius to the circle without using trig.

 

[willem2]

It's only possible to give an exact value of $\sin{\frac{\pi}{n}}$ if n is the product of a power of two, (0 or 1 powers) of 3, and an number of distinct fermat primes.

A fermat prime is a prime of the form 2^(2^k)+1

The only ones known are 5, 17,257,65537. Other values of n do not have an exact solutions. If n has a factor of 17,257 or 65537 the expressions will be EXTREMELY long.

 

[JungleJesus]

For any circle, given an angle (a) in radians, the arc-length (s) is related to (a) and the radius (r) by:
s = r*a

On the unit circle, r = 1, so
s = a

If I were to start at the point P(1, 0) and travel on an arc with a radius of 1 a distance of (s) counter-clockwise, how could I find the endpoint of the arc E(x,y) by using the circle equation x² + y² = 1 and the arc-length (s) without using the trig functions.

I can't use the trig functions to find the point because I will use the point to define Sin(a) and Cos(a). I must find the point through an independent process to define Cos and Sin.
In doing so, any angle converted into radians could be symbolically analyzed, rather than approximated.