Question

If `sin alpha = 4/5` and `alpha` lies in quadrant II, `cos beta = 5/13` and `beta` lies in quadrant I, what is `sin(alpha - beta)`?

Answer 1

`sin(alpha-beta)=56/65`

Explanation:

As `alpha` lies in QuDRnt II (note that `cosalpha` will be negative) and `sinalpha=4/5`,

`cosalpha=sqrt(1-sin^2alpha)=sqrt(1-(4/5)^2)=sqrt(1-16/25)=sqrt(9/25)=-3/5`

Further, as `beta` lies in quadrantnt I (note that `sinbeta` will be positive) and as `cosbeta=5/13`,

`sinbeta=sqrt(1-cos^2beta)=sqrt(1-(5/13)^2)=sqrt(1-25/169)=sqrt(144/169)=12/13`

Hence, `sin(alpha-beta)=sinalphacosbeta-cosalphasinbeta`

= `(4/5)xx(5/13)-(-3/5)(12/13)=20/65+36/65=56/65`