Question

Show that $f\left(x\right)=\sin x$ is an increasing function on $(-\frac{\pi }{2},\frac{\pi }{2})$

Answer 1

$f\left(x\right)=\sin x$

for the function be increasing function

$f^{\prime }\left(x\right)>0$

$f^{\prime }\left(x\right)=\cos x$

Now $-\frac{\pi }{2}<x<\frac{\pi }{2}$

which gives $\cos x>0$

$f^{\prime }\left(x\right)>0$

so $f\left(x\right)$ is increasing on $[-\frac{\pi }{2},\frac{\pi }{2}]$.