Q.From the curve $y=\sin x$, graph the functions

Step 1 (i) $y=\sin(-x)$. Since sine is an ODD function, $\sin(-x)=-\sin x$. So this is $y=-\sin x=-f(x)$: the reflection of $y=\sin x$ about the $x$-axis. (It also equals $f(-x)$, the $y$-axis reflection -- for an odd function these two reflections coincide.)

Step 2 (ii) $y=-\sin(-x)$. Using oddness again: $-\sin(-x)=-(-\sin x)=\sin x$. So this simplifies back to the ORIGINAL curve $y=\sin x$ unchanged -- the two sign flips cancel each other.

Step 3 (iii) $y=\sin\left(\dfrac\pi2+x\right)$. This is $y=f(x+\pi/2)$ -- a LEFT shift of $y=\sin x$ by $\pi/2$. The identity $\sin\left(\dfrac\pi2+x\right)=\cos x$ confirms the shifted sine curve traces out exactly the cosine curve. …

Simplify each expression using the oddness of $\sin x$ and the standard co-function identiti …

• Missing that (iii) and (iv) reach the SAME final curve ($\cos x$) even though they look like different transformation recipes …