Question

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

Answer 1

`sin x + cos x = sin x + sin (90 - x)`
Use trig identity:
`sin a + sin b = 2sin ((a + b)/2)cos ((a - b)/2)`
In this case:
`sin ((a + b)/2) = sin (45) = sqrt2/2`
`cos ((a - b)/2) = cos (45 - x) = sin (90 - (45 - x)) = sin (x + 45)`
Therefor,
`sin x + cos x = sqrt2sin(x + 45)`

Answer 2

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
`sin(x+ϕ)=sin(x)cos(ϕ)+cos(x)sin(ϕ)`,
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,
`sin(x+π/2)=cos(x)`.
Likewise, For `π/4`, the terms get the same amplitude,
`sin(x+π/4)=1/sqrt (2)(sin(x)+cos(x))`.
Rearranging the terms you be your desired answer.