What is sin⁡(x)+cos⁡(x) in terms of sine?

Salvatore Boone

Salvatore Boone

Answered

2021-12-27

What is sin(x)+cos(x) in terms of sine?

Answer & Explanation

Bubich13

Bubich13

Expert

2021-12-28Added 36 answers

Explanation:
Using Pythagorean Identity
sin2x+cos2x=1,  so  cos2x=1sin2x
cosx=±1sin2x
sinx+cosx=sinx±1sin2x
Using complement / cofunction identity
cosx=sin(π2x)
sinx+cosx=sinx+sin(π22)
lovagwb

lovagwb

Expert

2021-12-29Added 50 answers

Explanation
Suppose that sinx+cosx=Rsin(x+α)
Then
sinx+cosx=Rsinxcosα+Rcosxsinα
=(Rcosα)sinx+(Rsinα)cosx
The coefficients of sinx and of cosx must be equal so
Rcosα=1
Rsinα=1
Squaring and adding, we get
R2cos2α+R2sin2α=2  so  R2(cos2α+sin2α)=2
R=2
And now
cosα=12
sinα=12
So  α=cos1(12)=π4
sinx+cosx=2sin(x+π4)
Vasquez

Vasquez

Expert

2022-01-08Added 457 answers

Explanation:
Using pythagorean identity,
sin2x+cos2x=1
So, cos2x=1sin2x
By taking square root on both the sides,
cosx+sinx=sin±1sin2x
Using complement or cofunction identity,
cosx=sin(π2x)
sinx+cosx=sinx+sin(π2x)
Thus, the expression for sinx+cosx in terms of sine is sinx+sin(π2x).

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