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
If you don't know how to solve equation
u ^ 2 - u - 2 = 0
In google type:
quadratic equation online
When you see list of results click on:
Free Online Quadratic Equation Solver:Solve by Quadratic Formula
When page be open in rectangle type:
u ^ 2 - u - 2 = 0
and click option: solve it
You will see solution step-by step
solve csc^2x - cscx + 9 =11
1 answer
1 answer