Number of zeros in cosine for one period

Question

Stupid I know, but for $ \cos(kx+\phi) $ is the number of zeros in the first period always two? Just unsure of myself.

Answer 1

You want to "prove" it that it is $2$. Assume $k, \phi > 0$, simply solve the equation: $\cos(kx+\phi) = 0\iff kx + \phi = \dfrac{(2n+1)\pi}{2}\iff x = \dfrac{(2n+1)\pi - 2\phi}{2k} \in \left[0, \dfrac{2\pi}{k}\right]\iff n = 0,1.$ This means there are $2$ zeros in the first period.

Answer 2

For any sine/cosine function, the number of zero crossings will always be two. If you managed to understand that for any period but the first one, just notice that all the periods are strictly the same, that's why they're called periods ;)

Btw, for a function defined on $\mathbb{R}$, what would first mean ?