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

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

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1/(sin(x))=2

\frac{1}{sin ( x )} = 2

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

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

\frac{1}{sin ( x )} = 2

Multiply both sides by

sin (x)

\frac{1}{sin ( x )} = 2

Multiply both sides by

sin (x)

\frac{1}{sin ( x )} sin (x) = 2 sin (x)

Simplify

1 = 2 sin (x)

1 = 2 sin (x)

Switch sides

2 sin (x) = 1

Divide both sides by

2

2 sin (x) = 1

Divide both sides by

2

\frac{2 sin ( x )}{2} = \frac{1}{2}

Simplify

sin (x) = \frac{1}{2}

sin (x) = \frac{1}{2}

Verify Solutions

Find undefined (singularity) points:

sin (x) = 0

Take the denominator(s) of

\frac{1}{sin ( x )}

and compare to zero

sin (x) = 0

The following points are undefined

sin (x) = 0

Combine undefined points with solutions:

sin (x) = \frac{1}{2}

General solutions for

sin (x) = \frac{1}{2}

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

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

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

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

1 - 2 sin (θ) = 0

cos (2 θ) + 3 cos (θ) = 4

0 = 2 cos (x) - 3 sin (x)

- cos (x) - sin (x) = 0

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