What is y=3sinx+4cosx in arccos?

1 Answer
Jul 13, 2018

The answer is =arccos(y/5)+arctan(3/4)

Explanation:

y=3sinx+4cosx

y=4cosx+3sinx

Write y=rcos(x-phi)

Then,

y=rcosxcosphi+rsinxsinphi

Comparing the 2 functions

{(rcosphi=4),(rsinphi=3):}

=>, r^2(cos^2phi+sin^2phi)=4^2+3^2=16+9=25

=>, r=sqrt25=5

Therefore,

tanphi=3/4

phi=arctan(3/4)

And

y=5cos(x-phi)

cos(x-phi)=y/5

arccos(y/5)=x-phi

x=arccos(y/5)+phi

x=arccos(y/5)+arctan(3/4)