Question

✓3 cosec thita+2=0 find principal solution of this equation
​

Answer 1

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