Question

If 3costheta+4sintheta=Asin(theta+alpha),then values of A and alpha are?

Answer 1

`A=5` and `alpha=tan^(-1)(3/4)`

Explanation:

`3costheta+4sintheta`

= `4sintheta+3costheta`

= `5(sinthetaxx4/5+costhetaxx3/5)`

= `5(sinthetacosalpha+costhetasinalpha)`, where `alpha=tan^(-1)(3/4)`

= `5sin(theta+alpha)`

Hence `A=5` and `alpha=tan^(-1)(3/4)`

Note: `-` When we have `asintheta+bcostheta`, we take `sqrt(a^2+b^2)` outside and then weare left with `a/sqrt(a^2+b^2)` as `cosalpha` and `b/sqrt(a^2+b^2)` as `sinalpha`. Note that then `sin^2alpha+cos^2alpha=1` and `tanalpha=b/a`. Here we have `a=4` and `b=3` and hence we have `sqrt(3^2+4^2)=5` outside and `cosalpha=4/5` and `sinalpha=3/5` and `tanalpha=3/4` and hence `alpha=tan^(-1)(3/4)`.