What is the general solution for 2sin(x)-3=-csc(x) ?

The general solution for 2sin(x)-3=-csc(x) is x= pi/6+2pin,x=(5pi)/6+2pin,x= pi/2+2pin

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2sin(x)-3=-csc(x)

2 sin (x) - 3 = - csc (x)

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{π}{2} + 2 πn

Solution steps

2 sin (x) - 3 = - csc (x)

Subtract

- csc (x)

from both sides

2 sin (x) - 3 + csc (x) = 0

Rewrite using trig identities

- 3 + csc (x) + \frac{2}{csc ( x )} = 0

Solve by substitution

csc (x) = 2, csc (x) = 1

csc (x) = 2 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

csc (x) = 1 : x = \frac{π}{2} + 2 πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{π}{2} + 2 πn

cos^{2} (x) + 1 = 4

1 = sin^{2} (x)

- 2 sin^{2} (x) - cos (x) = - 1

tan^{2} (x) = 1 + sec (x)

8 sin^{2} (x) = 7

What is the general solution for 2sin(x)-3=-csc(x) ?
The general solution for 2sin(x)-3=-csc(x) is x= pi/6+2pin,x=(5pi)/6+2pin,x= pi/2+2pin