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)`.

Options

A. `(23)/(17)((sqrt(3)-1)/(2)+(1)/(sqrt(2)))`
B. `(23)/(17)((sqrt(3)+1)/(2)+(1)/(sqrt(2)))`
C. `(23)/(17)((sqrt(3)-1)/(2)-(1)/(sqrt(2)))`
D. `(23)/(17)((sqrt(3)+1)/(2)-(1)/(sqrt(2)))`

Answer

A

Explanation

Since , `costheta=(8)/(17)and0ltthetalt(pi)/(2)`
`rArrsinthetasqrt(1-(8^(2))/(17^(2)))=(15)/(17)`
The value of the given expression
`=cos30^(@)*costheta-sin30^(@)sintheta+cos45^(@)costheta+sin45^(@)sintheta+cos120^(@)costheta+sin120^(@)sintheta`
`=costheta((sqrt(3))/(2)+(1)/(sqrt(2))-(1)/(2))-sintheta((1)/(2)-(1)/(sqrt(2))-(sqrt(3))/(2))`
`=(8)/(17)((sqrt(3))/(2)+(1)/(sqrt(2))-(1)/(2))+(15)/(17)((sqrt(3))/(2)+(1)/(sqrt(2))-(1)/(2))`
`=(23)/(17)((sqrt(3)-1)/(2)+(1)/(sqrt(2)))`