<s>[INST] After what interval in degrees or radians do sine, cosine and tangent values repeat? Between $0$ to $2π$, I have noticed that $\sin x$, $\cos x$ and $\tan x$ values repeat for different values of $x$.
For example, $\sin 30 = \sin 150$
What exactly is the interval between two successive values of $x$ such that the value of $\sin x$, $\cos x$ or $\tan x$ are equal?
 [/INST] $1:$
Let $\sin x=\sin y$
Applying $\sin C-\sin D=2\sin\frac{C-D}2\cos\frac{C+D}2,$
we have  $2\sin\frac{x-y}2\cos\frac{x+y}2=0$
If $\sin\frac{x-y}2=0, \frac{x-y}2=n180^\circ\implies x-y=n180^\circ$ where $n$ is any integer
If $\cos\frac{x+y}2=0, \frac{x+y}2=(2m+1)90^\circ\implies x+y=(2m+1)180^\circ$ where $m$ is any integer
$2:$
Apply $\cos C-\cos D=-2\sin\frac{C-D}2\sin\frac{C+D}2,$
$3:$
$\displaystyle\tan x=\tan y\iff \frac{\sin x}{\cos x}=\frac{\sin y}{\cos y}$
$$\implies \sin x\cos y-\cos x\sin y=0\implies \sin(x-y)=0$$
$\implies x-y=r180^\circ$ where $r$ is any integer
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