How to turn a sine function into a cosine function? if possible could someone provide an example with steps? !

Several ways may be used to turn a sine function into a cosine function, using the fundamental properties of both trigonometric functions. Here I deal with two of them.

One way is using the property cos(x) = sin (90° - x). Other is using the identity sin² (x) + cos² (x) = 1.

You will find a detailed explanation and an example below.

Step-by-step explanation:

Form 1:

By the definition of the sine and cosine functions, sin (90° - x) = cos (x), and cos (90° - x) = sin (x).

Hence, starting with the basic function y = sin (x), you can convert it into a cosine function substituting sin(x) with cos (90 - x), obtaining:

y = cos (90 - x)

Now see an example:Turn the sine function f(x) = 3 sin (30° - 2x) into a cosine function.

f(x) = 3 sin (30° - 2x) = 3 cos [90° - (30° - 2x) ] = 3 cos [90° - 30° + 2x] =

      = 3 cos [60° + 2x ]

Form 2:

Also, you can use the fundamental identity: sin² (x) + cos² (x) = 1

From which: sin² (x) = 1 - cos²(x) ⇒ sin (x) = +/- √ [1 - cos² (x) ]

The same example:Turn the sine function f(x) = 3 sin (30° - 2x) into a cosine function.f(x) = 3 sin (30° - 2x) = +/- √[1 - 3 cos² (30° - 2x)]

1/8 1/16 1/32 should be the next numbers because they are dividing by two