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
Nghi N
May 13, 2018

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)

Hariharan Baskarraj
May 13, 2018

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.

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