Question

Solve on the interval [0,2pi): 2cscx+5=1

Answer 1

`x\in[0,\ 2\pi)\\\\2\csc x+5=1\ \ \ \ |-5\\\\2\csc x=-4\ \ \ \ |:2\\\\\csc x=-2\\\\(1)/(\sin x)=-2\to\sin x=-(1)/(2)\to x=-(\pi)/(6)+2k\pi\ \vee\ x=(7\pi)/(6)+2k\pi\\\\x\in[0,\ 2\pi)\to x=-(\pi)/(6)+2\pi=(11\pi)/(6)\ \vee\ x=(7\pi)/(6)\\\\Answer:\ \boxed{D.\ (7\pi)/(6),\ (11\pi)/(6)}}`

Answer 2

Answer: D.
`(7\pi )/(6) ,(11\pi )/(6)`

Explanation:

Isolating the function:

`2csc(x)+5=1\\2csc(x)=1-5\\csc(x)=(-4)/(2) \\csc(x)=-2`

Since
`csc(x)=(1)/(sin(x))` then:

`(1)/(sin(x)) =-2\\-(1)/(2)=sin(x)`

The angles in the interval [0,2
`\pi`] that comply with this are
`(7\pi )/(6) and (11\pi )/(6)`

`sin((7\pi )/(6))=-(1)/(2) \\sin((11\pi )/(6))=-(1)/(2) \\`