What is the general solution for csc^2(x)+3csc(x)-4=0 ?

The general solution for csc^2(x)+3csc(x)-4=0 is x= pi/2+2pin,x=-0.25268…+2pin,x=pi+0.25268…+2pin

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csc^2(x)+3csc(x)-4=0

csc^{2} (x) + 3 csc (x) - 4 = 0

x = \frac{π}{2} + 2 πn, x = - 0.25268 \dots + 2 πn, x = π + 0.25268 \dots + 2 πn

Solution steps

csc^{2} (x) + 3 csc (x) - 4 = 0

Solve by substitution

csc (x) = 1, csc (x) = - 4

csc (x) = 1 : x = \frac{π}{2} + 2 πn

csc (x) = - 4 : x = arccsc (- 4) + 2 πn, x = π + arccsc (4) + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = arccsc (- 4) + 2 πn, x = π + arccsc (4) + 2 πn

Show solutions in decimal form

x = \frac{π}{2} + 2 πn, x = - 0.25268 \dots + 2 πn, x = π + 0.25268 \dots + 2 πn

2 sin (θ) = - 3, 0 \leq θ < 2 π

cos^{2} (φ) = sin^{2} (φ)

sin^{2} (x) - 2 cos^{2} (x) = 0

sin (x) = \frac{1}{10}

(sin (θ) - 1) (sin (θ) + 1) = 0

What is the general solution for csc^2(x)+3csc(x)-4=0 ?
The general solution for csc^2(x)+3csc(x)-4=0 is x= pi/2+2pin,x=-0.25268…+2pin,x=pi+0.25268…+2pin