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