Question

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

Answer 1

We can convert sin function into cosine by adding Pi/2 or 90° to the x in sin(x) to get cosine.

Explanation:

sine function can be changed to cosine and vice versa by adding 90 degrees and its multiples in domain of function so

For instance

Sin (a+90)= cos a it is +ve as in angle lies in 2nd quad if a is less than 90 and sine is + ve in 2nd quad

Answer 2

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.

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)]