# Please could you explain how sin(x) + cos(x) = sqrt(2)sin(x+45)? I understand how you get sqrt(2) through sin(x)cos(45)+cos(x)sin(45). But I don’t. Understand how you end up with sin(x+45) or sin(x+π/4) depending on how you see it. Thanks a lot

##### 2 Answers

Use trig identity:

In this case:

Therefor,

Phase shift.

#### Explanation:

The sine and cosine are two facets of the same function, and morph into each other when you apply a "phase shift": by the addition formula

A shifted sine is a linear combination of a sine and a cosine. For specific values of the shift, one of the terms vanishes. For example,

Likewise, For

Rearranging the terms you be your desired answer.