I want to find out the angle for the expression a 3 + b 3...

Step 1
There is no single angle corresponding to the relationship $a^3+b^3=c^3$.
Suppose that a triangle has sides of lengths a,b, and c such that $a^3+b^3=c^3$. We know from the law of cosines that if θ is the angle opposite the side of length c, then $c^2=a^2+b^2-2ab\cos \theta$, so $\cos \theta =\frac{a^2+b^2-c^2}{2ab}.$.
Now let’s look at just a few examples. If $a=b=1$, then $c=\sqrt[3]{2}$, and $\cos \theta =\frac{2-2^{2/3}}{2}\approx 0.20630.$.
Step 2
If $a=1$ and $b=2$, then $c=\sqrt[3]{9}$, and $\cos \theta =\frac{5-9^{2/3}}{4}\approx 0.16831.$.
If $a=1$ and $b=3$, then $c=\sqrt[3]{28}$, and $\cos \theta =\frac{10-{28}^{2/3}}{6}\approx 0.12985.$.
As you can see, these values of $\cos \theta$ are all different, so the angles themselves are also different.

Step 1
You do get slightly different answers for the Pythagorean Theorem on the surface of a sphere of radius 1, or the hyperbolic plane of curvature -1. On the sphere, a right triangle with geodesic lengths of legs a,b and hypotenuse c obeys $\cos c=\cos a\cos b,$, while in the hyperbolic plane with curvature -1 it becomes $\cosh c=\cosh a\cosh b.$.
Step 2
In both cases, if you write out the functions as power series in a, b, c, you see that the limit as a, b, c all shrink to nearly 0 is the traditional Pythagorean Theorem.