The expressions 2cos x and cos 2x represent different mathematical concepts and should not be confused with one another.
2cos x is simply two times the cosine of angle x. This means that whatever the cosine value of x is, you just multiply that result by 2. For example, if x is 0 degrees (or 0 radians), then cos 0 is 1, making 2cos 0 equal to 2.
On the other hand, cos 2x uses the double angle identity for cosine. This expression computes the cosine of double the angle x itself. Using the double angle formula, we can also express it as cos 2x = cos² x – sin² x or cos 2x = 2cos² x – 1 or cos 2x = 1 – 2sin² x depending on what form is more useful for a particular calculation.
In summary:
- 2cos x scales the cosine value, effectively stretching the range of the cosine function.
- cos 2x changes the angle itself, affecting the periodic behavior and resulting values based on the properties of the cosine function.