Yesterday I posted a tweet containing a closed-form algebraic expression for the cosine of 1°. One might wonder how one can arrive at this formula. The answer is that one has to use the formulas of trigonometry for combining angles, or the formulas for the product of complex numbers or alternatively formulas for successive rotations. We'll start by giving a few formulas for rotations. The following matrix rotates a point on the x,y-plane an angle counter clockwise through an angle θ.
A number of useful formulas are the following in which a product is equated to the identity matrix which corresponds to a complete rotation of 360 degrees or another matrix for some other known rotation.
Notice that one can use the cube of a rotation matrix to get a cubic formula in reduced form involving only the cosine of the unknown angle and that of the known angle. One can use the Cardano formula to solve for κ. The last formula is for the difference between two known angles. These formulas can be used to find the sines and cosines of the following angles.
The solution of the first four matrix power formulas is simplified by using the components equal to zero and eliminating unneeded factors. One has to be careful about the κ in the fourth power equation since the solution is κ=0 for θ=90°. A check shows that the formulas used in the tweet work as desired.
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