# Thread: How does the range of the cos function change with exponents?

1. ## How does the range of the cos function change with exponents?

For example, what is the range of f(x) = cos^3 x? ...cos^2 x?

I understand the range of cosx and sinx but I'm not sure how to adjust for the exponents. I think there must be something simple that I'm just missing, so if someone could explain, that would be great.

Thanks.

2. ## Re: How does the range of the cos function change with exponents?

$\cos^n(x) = (\cos(x))^n$ for positive n.

In essence this means you take the range of cos(x) and raise both sides to the exponent n. Be wary of even exponents and negative numbers

3. ## Re: How does the range of the cos function change with exponents?

Hint: The range will be different for even powers of cos vs the odd powers. Does this help?

4. ## Re: How does the range of the cos function change with exponents?

Originally Posted by dsr430
For example, what is the range of f(x) = cos^3 x? ...cos^2 x?
If $-1\le A\le 1$ then $-1\le A^3\le 1~\&~0\le A^2\le 1.$

5. ## Re: How does the range of the cos function change with exponents?

So the range is [-1, 1] when the exponent is odd and [0, 1] when the exp is even?

6. ## Re: How does the range of the cos function change with exponents?

Originally Posted by dsr430
So the range is [-1, 1] when the exponent is odd and [0, 1] when the exp is even?
This is correct for positive integral exponents.