# definition of sinusoidal curve

I have question related with these two definition: In geometry, the sinusoidal spirals are a family of curves defined by the equation in polar coordinates

$$r^n = a^n \cos(n \theta)$$

where $a$ is a nonzero constant and $n$ is a rational number other than $0$. On the other hand, with a rotation about the origin, this can also be written

$$r^n = a^n \sin(n \theta)$$

Why so different? Rotation about the origin does create in some situation cosine and sine function same or? Please explain to me.

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A sinusoid is a function which can be written in the form $f(x) = R\sin (ax + b)$. So for example $\cos x = \sin (-x + \frac{\pi}{2})$, and so forth.