Find the exact value of each of the following under the give

Find the exact value of each of the following under the given conditions below. tan alpha = - 12/5, pi/2

Solution

given

tana =-12/5 , pi/2< a<pi

tana=opposite side/adjacent side

hypotenuse²=opposite side²+ adjacent side²

hypotenuse²=12²+ 5²

hypotenuse=13

cosa=adjacent side/hypotenuse

cosa=-5/13

sina=opposite side/hypotenuse

sina=12/13

given cosb=3/2 ,0<b<pi/2

cosb=adjacent side/hypotenuse

hypotenuse²=opposite side²+ adjacent side²

2²=opposite side²+ 3²

4=opposite side²+ 3

opposite side² =1

opposite side =1

sinb=opposite side/hypotenuse

sinb=1/2

tanb=opposite side/adjacent side

tanb=1/3

so sina=12/13,sinb=1/2,cosa=-5/13,cosb=3 /2,tana =-12/5,tanb=1/3

a)sin(a+b)=sinacosb +cosasinb

sin(a+b)=((12/13)(3 /2))+((-5/13)(1/2))

sin(a+b)=(123 -5)_/26

b)cos(a+b)=cosacosb -sinasinb

cos(a+b)=((-5/13)(3 /2))-((12/13)(1/2))

cos(a+b)=-(53 +12)_/26

c)

sin(a-b)=sinacosb -cosasinb

sin(a-b)=((12/13)(3 /2))-((-5/13)(1/2))

sin(a-b)=(123 +5)_/26

cos(a-b)=cosacosb +sinasinb

cos(a-b)=((-5/13)(3 /2))+((12/13)(1/2))

cos(a-b)=(-53 +12)_/26

d)tan(a-b)=sin(a-b)/cos(a-b)

tan(a-b)=[(123 +5)_/26]/[(-53 +12)_/26]

tan(a-b)=(123 +5)_{/(-53 +12)}