What is the general solution for 3sin(2x)=-2 ?

The general solution for 3sin(2x)=-2 is x=-(0.72972…)/2+pin,x= pi/2+(0.72972…)/2+pin

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3sin(2x)=-2

3 sin (2 x) = - 2

x = - \frac{0.72972 \dots}{2} + πn, x = \frac{π}{2} + \frac{0.72972 \dots}{2} + πn

Degrees

x = - 20.90515 \dots^{\circ} + 18 0^{\circ} n, x = 110.90515 \dots^{\circ} + 18 0^{\circ} n

Solution steps

3 sin (2 x) = - 2

Divide both sides by

3

3 sin (2 x) = - 2

Divide both sides by

3

\frac{3 sin ( 2 x )}{3} = \frac{- 2}{3}

Simplify

sin (2 x) = - \frac{2}{3}

sin (2 x) = - \frac{2}{3}

Apply trig inverse properties

sin (2 x) = - \frac{2}{3}

General solutions for

sin (2 x) = - \frac{2}{3}

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

2 x = arcsin (- \frac{2}{3}) + 2 πn, 2 x = π + arcsin (\frac{2}{3}) + 2 πn

2 x = arcsin (- \frac{2}{3}) + 2 πn, 2 x = π + arcsin (\frac{2}{3}) + 2 πn

Solve

2 x = arcsin (- \frac{2}{3}) + 2 πn : x = - \frac{arcsin ( \frac{2}{3} )}{2} + πn

2 x = arcsin (- \frac{2}{3}) + 2 πn

Simplify

arcsin (- \frac{2}{3}) + 2 πn : - arcsin (\frac{2}{3}) + 2 πn

arcsin (- \frac{2}{3}) + 2 πn

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{2}{3}) = - arcsin (\frac{2}{3})

= - arcsin (\frac{2}{3}) + 2 πn

2 x = - arcsin (\frac{2}{3}) + 2 πn

Divide both sides by

2

2 x = - arcsin (\frac{2}{3}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = - \frac{arcsin ( \frac{2}{3} )}{2} + \frac{2 πn}{2}

Simplify

x = - \frac{arcsin ( \frac{2}{3} )}{2} + πn

x = - \frac{arcsin ( \frac{2}{3} )}{2} + πn

Solve

2 x = π + arcsin (\frac{2}{3}) + 2 πn : x = \frac{π}{2} + \frac{arcsin ( \frac{2}{3} )}{2} + πn

2 x = π + arcsin (\frac{2}{3}) + 2 πn

Divide both sides by

2

2 x = π + arcsin (\frac{2}{3}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{π}{2} + \frac{arcsin ( \frac{2}{3} )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{π}{2} + \frac{arcsin ( \frac{2}{3} )}{2} + πn

x = \frac{π}{2} + \frac{arcsin ( \frac{2}{3} )}{2} + πn

x = - \frac{arcsin ( \frac{2}{3} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{2}{3} )}{2} + πn

Show solutions in decimal form

x = - \frac{0.72972 \dots}{2} + πn, x = \frac{π}{2} + \frac{0.72972 \dots}{2} + πn

x, arctan (y) = 2 arctan (x)

sin (2 x) = - \frac{3}{2}

tan (x) = \frac{7.3}{6.8}

cot (θ) = \frac{13}{6}

cos (θ) = \frac{2}{6}

What is the general solution for 3sin(2x)=-2 ?
The general solution for 3sin(2x)=-2 is x=-(0.72972…)/2+pin,x= pi/2+(0.72972…)/2+pin