I will quickly explain what I am expecting from this question. Given a point $p=(x,y)$ in the plane, we can look at the ratios $$A_1(p)=\dfrac{x}{r},\qquad A_2(p)=\dfrac{y}{r},$$ where $r$ is the distance form the origin to $p.$ There are two advantages of rewriting usual trigonometric functions in this weird way. First is we can develop trigonometry, without directly referring to angles, just based on ratios. In this new language, we can still derive all the trig-identities such as $$A_1(p)^2+A_2(p)^2=1,\qquad A_2(p+q)=A_2(p)A_1(q)+A_1(p)A_2(q),\qquad\text{etc.}$$ Here we can define $p+q,$ with no use of angels, as a counterclockwise rotation of $xy$-plane so that radial line of $p$ coincide of $x$-axis of the frame that contains $q.$ Since this "definition of addition" is asymmetric with respect to $y$-axis the formula for $A_1(p+q)$ won't be just a symbol pushing around but contain some useful information.
The second reason is this gives a straightforward generalization of trigonometric functions to higher dimensions. For example when $p=(x,y,z)$ in the $3$-space, we can define same ratios making necessary alterations: $$A_1=\dfrac{x}{r},\qquad A_2(p)=\dfrac{y}{r},\qquad A_3=\dfrac{z}{r}.$$ Then Pythagorean theorem says, $$A_1(p)^2+A_2(p)^2+A_3(p)^2=1.$$ Now, I am wondering about the "angle addition" in solid geometry.
- What would be a geometrically meaningful way of adding angles in $3$-space?
- What are the formulas of $A_1(p+q), A_2(p+q)$ and $A_3(p+q)$ under that addition?