Question

write y=sinx-cosx in the form y=ksin (x+a), where the measure of a is in radians.

Answer 1

The coefficients of sin(x) and -cos(x) are 1 and -1 respectively.
We divide each term by the factor sqrt(1²+(-1)²)=sqrt(2)
y=sinx-cosx
=>
y=sqrt(2)[sin(x)*(1/sqrt(2)+cos(x)*(-1/sqrt(2))

Since
cos(-π/4)=1/sqrt(2), and
sin(-π/4)=-1/sqrt(2),
we set a=3π/2, then we can rewrite the above as:
y=sqrt(2)*[sin(x)cos(a)+cos(x)sin(a)]
=sqrt(2)sin(x+a)
where k=sqrt(2) and a=-π/4