1.

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

Answer»

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

Solution :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`
WHICHIS an odd prime.


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