What is the general solution for 1/(sin(x))=sqrt(5) ?

The general solution for 1/(sin(x))=sqrt(5) is x=0.46364…+2pin,x=pi-0.46364…+2pin

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1/(sin(x))=sqrt(5)

\frac{1}{sin ( x )} = 5

x = 0.46364 \dots + 2 πn, x = π - 0.46364 \dots + 2 πn

Solution steps

\frac{1}{sin ( x )} = 5

Multiply both sides by

sin (x)

1 = 5 sin (x)

Switch sides

5 sin (x) = 1

Divide both sides by

5

sin (x) = \frac{5}{5}

Verify Solutions

Find undefined (singularity) points:

sin (x) = 0

Combine undefined points with solutions:

sin (x) = \frac{5}{5}

Apply trig inverse properties

x = arcsin (\frac{5}{5}) + 2 πn, x = π - arcsin (\frac{5}{5}) + 2 πn

Show solutions in decimal form

x = 0.46364 \dots + 2 πn, x = π - 0.46364 \dots + 2 πn

tan (2 x) + tan (4 x) = 0

2 sin (θ) - 3 = 0, 0 \leq θ < 2 π

arctan (θ) = - 1

tan^{2} (θ) - tan (θ) - 2 = 0

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

What is the general solution for 1/(sin(x))=sqrt(5) ?
The general solution for 1/(sin(x))=sqrt(5) is x=0.46364…+2pin,x=pi-0.46364…+2pin