Degrees or radian in Euler's identity?

Question

In Euler's identity, should values in the RHS (the sin and cos part) always be in radians? like $e$ to the power of imaginary $\pi$ is negative one, so if I had $e$ to just the power of $i$, then I would take the radian values of $\cos(1)$ and $\sin(1)$ right?

Answer 1

This is a good question, because the answer gets to the heart of what an angle actually is. There are many ways to formally define this concept, but the simplest is that an angle denotes the ratio of an arc length to a radius. This idea is particularly important in the context of unit circle. Since the unit circle has a radius of $1$, $\cos\theta$ and $\sin\theta$ correspond to the $x-$ and $y$-coordinates that you arrive at after tracing an arc of $\theta$ units counterclockwise around, where your starting position is $(1,0)$. When angles are understood in this way, the question of how they are measured becomes rather odd. An angle is a dimensionless quantity, a pure number that represents a ratio. The degrees symbol is a bizarre scale factor of $\pi/180$. For instance, $$ \sin(90^{\circ})=\sin(90 \cdot \pi/180)=\sin(\pi/2)=1 \, . $$ Euler's formula, $e^{it}=\cos t+i\sin t$ has a very simple geometric interpretation. If a particle is moving around the unit circle centred at the origin of the complex plane, then at time $t=0$, the particle has position $(1,0)$. The particle then moves at unit speed anticlockwise around the unit circle, so that at time $t$ the particle has moved $t$ units along the circle's arc. Hence, by the geometric definitions of sine and cosine, the position of the particle is $(\cos t,\sin t)$.