Question

write sin(x)+2cos(x) in terms of sine only

Answer 1

Take a look at the graphs of sin(x) and cos(x): http://puu.sh/8nz0m.png

You should notice that cos(x) is just sin(x) shifted to the left pi/2 units.

So, we want to shift cos(x) to the right pi/2 units.

Since we're shifting to the right, we need to account for this inside the argument of cos(x).

cos(x-pi/2) = sin(x). That's the relationship between these two functions.

But we're writing cos(x) in terms of sin(x). So, we should shift sin(x) to the left pi/2 units.

Here is the other relationship: sin(x+pi/2) = cos(x).

So now, sin(x) + 2cos(x) = sin(x) + 2(sin(x+pi/2)).