## anonymous one year ago I will fan and medal! Why is cosnπ = (–1)^n true, when n could be any integer?

1. anonymous

Let's think about this systematically. If n can be an integer, then n can be 1, 2, 3, 0, -1, -2, -3, etc. Look at what happens with $$\cos(n\pi)$$ with these values. No matter which integer value of n it is, $$n\pi$$ will have a factor of $$\pi$$ in it. The value of cosine when the angle is at a value like 0, $$\pi$$, $$2\pi$$, $$3\pi$$, $$-\pi$$, $$-2\pi$$, etc. are either 1 or -1. Cosine would evaluate to 1 when n is an even integer (because then the angle would have factors of $$2\pi$$) and -1 when n is an odd number. Likewise, on the right side, $$(-1)^n$$ evaluates to -1 when n is odd and to 1 when n is even. Because the right side equals the left at the same integer values of n, they are equal.

2. anonymous

Does this make sense or should I try and clarify further?

3. anonymous

no i understand. thank you @Calcmathlete

4. anonymous

np :)