If `cottheta=5//12andtheta` lies in the third quadrant , then what is `(2sintheta+3costheta)` equal to?

Options

A. `-4`
B. `-p^(2)` for some odd prime p
C. `(-q//p)` where p is an odd prime and q a positive integer with `(q//p)` not an integer
D. `-p` for some odd prime p

Answer

D

Explanation

Let `cottheta=(5)/(12)`
`rArrtantheta=(12)/(5)=("perpendicular"(P))/("base"(b))`
`therefore"Hypotenuse"(H)=sqrt(p^(2)+b^(2))`
`=sqrt((12)^(2)+(5)^(2))=sqrt(144+255)=sqrt(169)=13`
Consider 2 `2sintheta+3costheta`
`=2((p)/(H))+3((b)/(H))` " " (H- Height)
But `theta` lies in `3^(rd)` quadrant and sin `theta , costheta` both are negative in `3^(rd)` quadrant
`therefore2sintheta+3costheta=2((-p)/(H))+3((-b)/(H))`
`=2((-12)/(13))+3((-5)/(13))`
`=(-24-15)/(13)=(-39)/(13)=-3`
which is an odd prime.