Question

solve csc^2x - cscx + 9 =11

Answer 1

csc ^ 2 ( x ) - csc ( x ) + 9 = 11 Subtract 11 to both sides

csc ^ 2 ( x ) - csc ( x ) + 9 - 11 = 11 - 11

csc ^ 2 ( x ) - csc ( x ) - 2 = 0


Substitute :

csc ( x ) = u


u ^ 2 - u - 2 = 0


The exact solutions are :

u = - 1

and

u = 2


u = csc ( x ) so :


csc ( x ) = - 1

and

csc ( x ) = 2



csc ( - pi / 2 ) = - 1

csc ( x ) is a periodic function with period 2 pi n

where n is an integer

Solutions :

x = 2 pi n - pi / 2

and

x = 2 pi n - pi / 2 + 2 pi =

2 pi n - pi / 2 + 4 pi / 2 =

2 pi n + 3 pi / 2



csc ( pi / 6 ) = 2

csc ( 5 pi / 6 ) = 2

csc ( x ) is a periodic function with period 2 pi n

where n is an integer

Solutions :

x = 2 pi n + pi / 6

and

x = 2 pi n + 5 pi / 6



Final solutions :


x = 2 pi n - pi / 2

x = 2 pi n + 3 pi / 2

x = 2 pi n + pi / 6

x = 2 pi n + 5 pi / 6



P.S

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u ^ 2 - u - 2 = 0

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u ^ 2 - u - 2 = 0

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