✓3 cosec thita+2=0 find principal solution of this equation
Answers
Answer:
We know that principal value branch of
\begin{gathered} {cosec}^{ - 1} x \:\: is\:\: [-\frac{\pi}{2}, \frac{\pi}{2}] \\ \end{gathered}
cosec
−1
xis[−
2
π
,
2
π
]
-{0}
To solve the given equation we must keep consider that cosec and cosec inverse cancels each other only if x belongs to its principal value .
\begin{gathered} \sqrt{3} cosec\: x + 2=0 \\ \\\sqrt{3} cosec\: x = - 2 \\ \\ cosec\: x = \frac{ - 2}{ \sqrt{3} } \\ \\ x = {cosec}^{ - 1} ( \frac{ - 2}{ \sqrt{3} } )\\ \\ we \: know \: that \: cosec \: \frac{\pi}{3} = \frac{2}{ \sqrt{3} } \\ \\We\:\: know\:\:that\:\: \\ \\ {cosec}^{ - 1} ( - cosec\: x) = - {cosec}^{ - 1} ( cosec\: x) \\\\ So\\\\ x= {cosec}^{ - 1} (cosec ( -\frac{\pi}{3} )) \\ \\x=- {cosec}^{ - 1} (cosec ( \frac{\pi}{3} ))\\\\ x= - \frac{ \pi}{3} \: \: \: \: (x \: belongs \: to \: [- \frac{\pi}{2}, \frac{\pi}{2}])-(0)\\ \\ x = \frac{ - \pi}{3} \\ \end{gathered}
3
cosecx+2=0
3
cosecx=−2
cosecx=
3
−2
x=cosec
−1
(
3
−2
)
weknowthatcosec
3
π
=
3
2
Weknowthat
cosec
−1
(−cosecx)=−cosec
−1
(cosecx)
So
x=cosec
−1
(cosec(−
3
π
))
x=−cosec
−1
(cosec(
3
π
))
x=−
3
π
(xbelongsto[−
2
π
,
2
π
])−(0)
x=
3
−π