What is the general solution for 2sin(x)+1=csc(x) ?

The general solution for 2sin(x)+1=csc(x) is x=(3pi)/2+2pin,x= pi/6+2pin,x=(5pi)/6+2pin

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2sin(x)+1=csc(x)

2 sin (x) + 1 = csc (x)

x = \frac{3 π}{2} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Solution steps

2 sin (x) + 1 = csc (x)

Subtract

csc (x)

from both sides

2 sin (x) + 1 - csc (x) = 0

Rewrite using trig identities

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

Solve by substitution

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

csc (x) = - 1 : x = \frac{3 π}{2} + 2 πn

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

Combine all the solutions

x = \frac{3 π}{2} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

tan^{2} (x) = - tan (x)

7 sin (x) + 3 = sin (x)

2 cos (2 x) = 3

cos (2 x) = 1 + 4 sin (x)

(tan (θ) + 1) (cos (θ) - 1) = 0

What is the general solution for 2sin(x)+1=csc(x) ?
The general solution for 2sin(x)+1=csc(x) is x=(3pi)/2+2pin,x= pi/6+2pin,x=(5pi)/6+2pin