graph of cosx + sinx in mathematica

Question

How do you get the graph of cosx + sinx in mathematica?

Answer 1

sin(x+¼π)=sin(x)cos(¼π)+cos(x)sin(¼π).

But sin(¼π)=cos(¼π), because sin(¼π)/cos(¼π)=tan(¼π)=1.

sin(¼π)=cos(¼π)=√2/2, so sin(x+¼π)=(√2/2)(sin(x)+cos(x)) or (sin(x)+cos(x))/√2.

This means that, if we plot √2sin(x+¼π) we get the graph of sin(x)+cos(x).

This further implies that we have a simple sine wave with a maximum value of √2 instead of 1 and a minimum value of -√2 instead of -1 (that is, amplitude=√2 instead of 1). And there's a phase shift of -¼π (that is, -45°) which pushes the sine wave to the left by one eighth of a cycle.