Question

For all integers n, sin(nx) = cos(?) for some ?
1) 0
2) 1
3) 2
4) 3

Answer 1

The value of ? for which sin(nx) = cos(?) is 1.

Step-by-step explanation:

In order for sin(nx) to be equal to cos(x), the argument of the cosine function must be an integral multiple of π/2. Therefore, ? = 1 is the correct answer. Let's go through the steps to understand why.

Using the identity sin(a + b) = sin(a)cos(b) + cos(a)sin(b), we can rewrite sin(nx) as sin((n-1)x + x). We can further expand this as sin((n-1)x)cos(x) + cos((n-1)x)sin(x).

Now, if we compare this to cos(x), we can see that they will be equal if sin((n-1)x)cos(x) = cos((n-1)x)sin(x). This condition is satisfied if the argument of the cosine is an integral multiple of π/2. Therefore, ? = 1 is the correct answer.