What is the general solution for solvefor y,x=4sin(2y) ?

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

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solvefor y,x=4sin(2y)

solve for

y, x = 4 sin (2 y)

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

Solution steps

x = 4 sin (2 y)

Switch sides

4 sin (2 y) = x

Divide both sides by

4

4 sin (2 y) = x

Divide both sides by

4

\frac{4 sin ( 2 y )}{4} = \frac{x}{4}

Simplify

sin (2 y) = \frac{x}{4}

sin (2 y) = \frac{x}{4}

Apply trig inverse properties

sin (2 y) = \frac{x}{4}

General solutions for

sin (2 y) = \frac{x}{4}

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

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

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

Solve

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

2 y = arcsin (\frac{x}{4}) + 2 πn

Divide both sides by

2

2 y = arcsin (\frac{x}{4}) + 2 πn

Divide both sides by

2

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

Simplify

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

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

Solve

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

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

Divide both sides by

2

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

Divide both sides by

2

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

Simplify

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

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

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

cos^{2} (x) - \frac{5}{2} cos (x) - \frac{3}{2} = 0

x, sin (4 x) = \frac{1}{2}

sec (2 x - 1 0^{\circ}) = csc (5 0^{\circ})

7 sec (x) - 7 = 0, 0 \leq x \leq 2 π

cos (x) + cos (x) = cos (2 x)

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