The Sine of 1 Degree

Note: I provide this tutorial with no guarentee of its accurracy. It is here for your own enjoyment. If you would like to use it in your own papers or theories, the work should be checked and a refrence to the author would be nice.

Have you ever wondered why your teacher only gives you the degrees of 30, 45 and 60 degrees? Did you ever think that others were possible? Well, you can use some identites and find all degrees of 15, this leaves us with 0, 15, 30, 45, 60, 75 and 90 with exact values known. So, are there others that are known?

Of course! But the key is not normally given in a high school class room for 2 reasons:

1. It involves some simple complex number analysis to find the Sin[72 Degree].

2. It involves the cubic equation.

The work is messy, but it can be done! The way I've chosen below is not the only way to do this, but its the way I did it and the simplist I've seen.

To start off, we need to know the exact value of Sin[75 Degree]. To do this, we use the the expansion Sin[x+y]==Cos[y] Sin[x]+Cos[x] Sin[y]. This yields:

Now, we need to find the Sin[72 Degree]. To find this, refer to this by Paul R. Patten. It involves taking the 5th root of 1+0*i, expanding and solving the resulting equations. The end result is this:

Now, we can use the identity Sin[x-y]==Cos[y] Sin[x]-Cos[x] Sin[y] fo find the exact value of the Sin[3 Degree]. The simplified result is below.

Ok, now we have the Sin[3 Degree]. We broke the multiple of 15 barrier, but we havn't broken the multiple of 3 barrier yet. Every multiple of 3 degrees can now be expressed in symbolic form with a little bit of work thanks to the Sin[72 Degree]. The final step involves generating a 1/3 degree formula. There exists a 1/2, 1/4 and 1/8 degree formula (all the latter generated from the first), but none other. To find this 1/3 formula, all we have to do is expand Sin[3*x] and solve for the Sin[x]. So, the Sin[3*x]==3*Sin[x]+4*Sin[x]^3 where we set x to be 1 Degree. Now that we know what the Sin[3 Degree] is, it is possible to solve for the Sin[1 Degree]. This is a cubic equation with a=-4, b=0, c=3 d= -Sin[3 Degree]. Plugging this into the cubic equation, we generate the solution we want (check your work numerically because 2 of the solutions are wrong). Hence, we get the result below (simplified):

Using Mathematica's FunctionExpand (4.1), I was able to produce a slightly more simplified equation:

There it is! The equation does involve complex numbers but they all equal out to zero. The only way to get rid of the complex numbers is to use triginometry and you will see that it reduces to the Sin[Pi/180] which is the same as Sin[Pi Degree].

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