Question

How do you simplify `sin(x-pi/2)`?

Answer 1

`-cosx`

Explanation:

Using the appropriate `color(blue)"addition formula"`

`color(orange)"Reminder"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(sin(A-B)=sinAcosB-cosAsinB)color(white)(a/a)|)))`

`rArrsin(x-pi/2)=sinxcos(pi/2)-cosxsin(pi/2)`

now `cos(pi/2)=0" and " sin(pi/2)=1`

`rArrsin(x-pi/2)=sinx(0)-cosx(1)=-cosx`

This is a useful result and worth knowing for future reference.

Answer 2

`-cosx`

Explanation:

The answer given prior is a perfectly valid explanation, but here is another:

We must consider our knowledge of transformations:

`f(x-alpha)` is a translation of `f(x)` by `(alpha,0)`

So hence:

`sin(x- pi/2)` is just `sinx` translated by `(pi/2,0)`

We see that `sinx`: graph{sinx [-4.006, 4.006, -2.003, 2.003]}

`therefore sin(x-pi/2)`: graph{sin(x- pi/2) [-4.006, 4.006, -2.003, 2.003]}

By observing this new graph, we see that this is just `cosx` reflected in the `x` axis

Hence just:

`=> -cosx`