How to make a cos function into a sin function

Question

I need to convert this equation into a sin function: f(x) = 12 cos(2x + 1) − 3 I know cos(x)= sin (pi/2 -x) but other than that I dont know how to solve this problem

Answer 1

No, that's it.

$f(x) = 12 \cos(2x + 1) − 3$ and $\cos(x)= \sin (\pi/2 -x)$

$\therefore f(x) = 12 \sin(\pi/2-1-2x)-3$

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Though you could also use: $\cos(A+B) = \cos(A)\cos(B) - \sin(A)\sin(B)$

... and: $\sin(A+B) = \sin(A)\cos(B)+\cos(A)\sin(B)$

It won't come out any neater.

Answer 2

You can use the following half angle formula: $$\cos (2y)=1-2\sin^2(y)$$

Is $f(x)$ a known function?

Answer 3

If you define $y = 2x + 1$ you can write $f(x)$ as: $$f(x) = 12 \cos(y) -3$$ If you now use that $\cos(y)=\sin(\pi/2-y)$ you get: $$ f(x)=12 \sin(\pi/2 - y) -3 $$ Using $y = 2x + 1$ you now get: $$ f(x)=12 \sin(\pi/2 - (2x +1)) -3 $$