How do you solve #sqrt3csc(9x)-7=-5#?

2 Answers
Mar 31, 2018

#x=pi/27, (2pi)/27, (7pi)/27, (8pi)/27,...#

Explanation:

#sqrt3csc(9x)-7=-5#

#sqrt3csc(9x)=7-5#

#sqrt3csc(9x)=2#

#csc(9x)=2/sqrt3#

#csc(pi/3)=2/sqrt3#

Comparing, the fundamental value of 9x satisfying the above condition is
#9x=pi/3#

#x=pi/27#
Further, the function cscx is positive in first and second quadrants

#9x=pi/3, pi-pi/3, 2pi+pi/3, 3pi-pi/3,....#

#9x=pi/3, (2pi)/3, (7pi)/3, (8pi)/3,...#

#x=pi/27, (2pi)/27, (7pi)/27, (8pi)/27#

Mar 31, 2018

#x=pi/27+(n2pi)/9# , # \ \ \ \x=(2pi)/27+(n2pi)/9#

Explanation:

#sqrt(3)csc(9x)-7=-5#

#csc(9x)=2/(sqrt(3)#

Identity:

#color(red)bb(csc(x)=1/sin(x)#

#1/(sin(9x))=2/(sqrt(3))#

#sin(9x)=(sqrt(3))/2#

#9x=arcsin(sin(9x))=arcsin((sqrt(3))/2)#

#9x=pi/3+n2pi# , # \ \ \ \9x=(2pi)/3+n2pi#

#x=pi/27+(n2pi)/9# , # \ \ \ \x=(2pi)/27+(n2pi)/9#

General solution:

For:

#n in ZZ#