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I would like to ask a question regarding the transformation of the graph $\cos(x)$ into $\sin(x)$ by means of transformations of graphs. Trigonometric Graphs

I used Desmos to render $3$ equations - $\cos(x)$, $\cos(-x)$ and $\cos(-x + \frac{\pi}{2})$

Now, when I transform $\cos(x)$ into $\cos(-x)$, this is a reflection in the $y$-axis, hence there is no change so the two graphs overlap.

However, when I add $\frac{\pi}{2}$, I expected the graph to shift left by $\frac{\pi}{2}$ but instead it shifts right by $\frac{\pi}{2}$ to give the sine function.

I might be missing something here, but why does this happen?

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    $\begingroup$ because you have $-x$, the shift is reversed. $\endgroup$ – Vasya Mar 22 '18 at 22:39
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Cosine is an even function; hence, $\cos(x)=\cos(-x)$. For $\cos(-x+\frac{\pi}{2})$, you can multiply the argument by $-1$ and not change the value of the function, which gives $\cos(x-\frac{\pi}{2})$. Now the transformation is clearly a shift to the right by $\frac\pi2$ which yields the sine function.

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To correctly determine to which direction the graph of any given function $f{(x)}$ shift upon a given transformation $f{(a+x)}$ or $f{(a-x)}$ ask yourself the following question.

"if $f{(x)}$ was $f{(0)}$ at $x=0$ for what value of $x$ will $f{(a-x)}$ be equal to $f{(0)}$."

The answer (i.e. $a$ here) will be the number you need to shift the graph to the right

(If the answer was "$-a$" you'd have to shift "$-a$" units to the right ,meaning, shifting "$a$" units to the left) . now you would have the transformed graph.

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