Question

If `sinalpha=3/5` and `alpha` lies in Q2 and `sinbeta=5/13` and `beta` lies in Q1, find `cos(alpha+beta)`, `cosbeta` and `tanalpha`?

Answer 1

`cos(alpha+beta)=-63/65`; `cosbeta=12/13` and `tanalpha=-3/4`

Explanation:

As `sinalpha=3/5` and `alpha` lies in Q2, `cosalpha` is negative and

`cosalpha=sqrt(1-sin^2alpha)=sqrt(1-(3/5)^2)`

= `sqrt(1-9/25)=sqrt(16/25)=-4/5`

and `tanalpha=sinalpha/cosalpha=(3/5)/(-4/5)=3/5xx(-5/4)=-3/4`

further as `sinbeta=5/13` and `beta` lies in Q1, `cosalpha` is positive and

`cosbeta=sqrt(1-sin^2beta)=sqrt(1-(5/13)^2)`

= `=sqrt(1-25/169)sqrt(144/25)=12/13`

and `cos(alpha+beta)`

= `cosalphacosbeta-sinalphasinbeta`

= `-4/5xx12/13-3/5xx5/13`

= `-48/65-15/65=-63/65`