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#?

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Answer 1 · 2017-03-16T16:16:09

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

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#