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

The general solution for solvefor y,x=sin(1/2 y) is y=2arcsin(x)+4pin,y=2pi+2arcsin(-x)+4pin

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solvefor y,x=sin(1/2 y)

solvefor

y = 2 arcsin (x) + 4 πn, y = 2 π + 2 arcsin (- x) + 4 πn

Solution steps

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

Switch sides

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

Apply trig inverse properties

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

General solutions for

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

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π + arcsin (a) + 2 πn

\frac{1}{2} y = arcsin (x) + 2 πn, \frac{1}{2} y = π + arcsin (- x) + 2 πn

\frac{1}{2} y = arcsin (x) + 2 πn, \frac{1}{2} y = π + arcsin (- x) + 2 πn

Solve

\frac{1}{2} y = arcsin (x) + 2 πn : y = 2 arcsin (x) + 4 πn

\frac{1}{2} y = arcsin (x) + 2 πn

Multiply both sides by

2

\frac{1}{2} y = arcsin (x) + 2 πn

Multiply both sides by

2

2 \cdot \frac{1}{2} y = 2 arcsin (x) + 2 \cdot 2 πn

Simplify

y = 2 arcsin (x) + 4 πn

y = 2 arcsin (x) + 4 πn

Solve

\frac{1}{2} y = π + arcsin (- x) + 2 πn : y = 2 π + 2 arcsin (- x) + 4 πn

\frac{1}{2} y = π + arcsin (- x) + 2 πn

Multiply both sides by

2

\frac{1}{2} y = π + arcsin (- x) + 2 πn

Multiply both sides by

2

2 \cdot \frac{1}{2} y = 2 π + 2 arcsin (- x) + 2 \cdot 2 πn

Simplify

y = 2 π + 2 arcsin (- x) + 4 πn

y = 2 π + 2 arcsin (- x) + 4 πn

y = 2 arcsin (x) + 4 πn, y = 2 π + 2 arcsin (- x) + 4 πn

What is the general solution for solvefor y,x=sin(1/2 y) ?
The general solution for solvefor y,x=sin(1/2 y) is y=2arcsin(x)+4pin,y=2pi+2arcsin(-x)+4pin