If `theta` lies in the first quardrant and `costheta=(8)/( 17 )`, then find the value of `cos(30^(@)+theta)+cos(45^(@)-theta)+cos(120^(@)-theta)`.

Explanation

Given that, `" "cos3theta=(8)/(17)rArrsintheta=sqrt(1-(64)/(289))`
`rArr" "sintheta=sqrt((289-64)/(289))rArrsintheta=pm(15)/(17)`
`rArr" "sintheta=(15)/(17)" "` [since, `theta` lies in the first quadrant]
Now, `cos(30^(@)+theta)+cos(45^(@)-theta)+cos(120^(@)-theta)`
`" "=cos(30^(@)+theta)+cos(45 ^(@)-theta)+cos(90^(@)+30^(@)-theta)`
`" "=cos(30^(@)+theta)+cos(45^(@) -theta)-sin(30^(@)-theta)`
`" "=cos30^(@)costheta-sin 30^(@) sin theta+cos45^(@) cos theta+sin45^(@) sintheta-sin30^(@) costheta+cos30^(@)sintheta`
`" "=(sqrt(3) )/(2)costheta-(1)/(2) sintheta+(1)/(sqrt(2))costheta+ (1)/(sqrt (2)) sintheta-(1)/(2)costheta(sqrt(3))/(2)sintheta`
`" "=((sqrt(3))/(2)+(1)/(sqrt(2))-(1)/(sqrt(2)))costheta+((1)/(sqrt(2))-(1)/(2)+(sqrt(3))/(2))sintheta`
`"" =((sqrt(6)+2-sqrt(2))/(2sqrt(2)))costheta+((2-sqrt(2)+sqrt(6))/(2sqrt(2)))sintheta`
`" "=((sqrt(6)+2-sqrt(2))/(2sqrt(2)))(8)/(17)+((2-sqrt(2)+sqrt(6))/(2sqrt(2)))(15)/(17)`
`" "=(1)/(17(2sqrt2))(8sqrt(6)+16-8sqrt(2)+30-15sqrt(2)+15sqrt(6)`
`" "=(1)/(17(2sqrt(2)))(23sqrt(6)-23sqrt(2)+46)`
`" "=(23sqrt(6))/(17(2sqrt(2)))-(23sqrt(2))/(17(2sqrt(2)))+(46)/(17(2sqrt2))`
`" "=(23sqrt(3))/(17(2))-(23)/(17(2))+(23)/(17sqrt(2))`
`" "=(23)/(17)((sqrt(3)-1)/(2)+(1)/(sqrt(2)))`