# Rising/lowering operators and trigonometric functions

I've just started learning about angular momentum and spin theory, and when I came across the definitions of the rising and lowering operators, I noticed the inverse form looks suspiciously like the definitions of $$\sin$$ and $$\cos$$ in terms of exponentials, is there a reason for why they are so similar?

$$l_x = \frac{l_+ + l_-}{2}, \ l_y = \frac{l_+ - l_-}{2i}$$

and $$\cos x = \text{Re}(e^{ix}) = \frac{e^{ix} + e^{-ix}}{2},$$ $$\sin x = \text{Im}(e^{ix}) = \frac{e^{ix} - e^{-ix}}{2i}.$$

• Please do not post formulae as screenshots, but use MathJax instead. – ZeroTheHero May 2 at 21:55

$$l_\pm$$ is defined as $$l_x \pm il_y$$. In general, any complex number $$z$$, $$\Re(z)=\dfrac{z+\bar z}{2}$$ and $$\Im(z)=\dfrac{z-\bar z}{2i}$$, which in the posted case is just the equations for $$l_{x,y}$$ and the trigonometric functions