The graphs are plots of functional forms $T^{-1}(T(x))$ where T is a trigonometric function:sine,cosine,tangent,cosecant,cotangent,and secant
Can someone please explain me indivially case by case why is the nature of the plots periodically changing in this manner,should'nt all of them be just straight lines? Please help me clarify this doubt.Also how would $T(T^{-1}(x))$ 's graphs be?
The point is that a trig functions is periodic, therefore not one-to-one. So the corresponding "inverse trig" function is only an inverse for the function on a certain interval. Thus $\arcsin(y)$ is the value $x$ in the interval $-\pi/2 \le x \le \pi/2$ such that $\sin(x)=y$. If $-\pi/2 \le x \le \pi/2$ we have $\arcsin(\sin(x)) = x$, but this can't be the case for $x$ outside that interval. As $x$ goes from $\pi/2$ to $3\pi/2$, $\sin(x)$ decreases from $1$ to $-1$, and thus $\arcsin(x)$ decreases from $\pi/2$ to $-\pi/2$. That gives you this part of the graph:
And then because $\sin$ is periodic with period $2\pi$, the pattern is repeated.