What is the general solution for sin(2x)=-1/3 ?

The general solution for sin(2x)=-1/3 is x=-(0.33983…)/2+pin,x= pi/2+(0.33983…)/2+pin

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sin(2x)=-1/3

sin (2 x) = - \frac{1}{3}

x = - \frac{0.33983 \dots}{2} + πn, x = \frac{π}{2} + \frac{0.33983 \dots}{2} + πn

Degrees

x = - 9.73561 \dots^{\circ} + 18 0^{\circ} n, x = 99.73561 \dots^{\circ} + 18 0^{\circ} n

Solution steps

sin (2 x) = - \frac{1}{3}

Apply trig inverse properties

sin (2 x) = - \frac{1}{3}

General solutions for

sin (2 x) = - \frac{1}{3}

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

2 x = arcsin (- \frac{1}{3}) + 2 πn, 2 x = π + arcsin (\frac{1}{3}) + 2 πn

2 x = arcsin (- \frac{1}{3}) + 2 πn, 2 x = π + arcsin (\frac{1}{3}) + 2 πn

Solve

2 x = arcsin (- \frac{1}{3}) + 2 πn : x = - \frac{arcsin ( \frac{1}{3} )}{2} + πn

2 x = arcsin (- \frac{1}{3}) + 2 πn

Simplify

arcsin (- \frac{1}{3}) + 2 πn : - arcsin (\frac{1}{3}) + 2 πn

arcsin (- \frac{1}{3}) + 2 πn

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{1}{3}) = - arcsin (\frac{1}{3})

= - arcsin (\frac{1}{3}) + 2 πn

2 x = - arcsin (\frac{1}{3}) + 2 πn

Divide both sides by

2

2 x = - arcsin (\frac{1}{3}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = - \frac{arcsin ( \frac{1}{3} )}{2} + \frac{2 πn}{2}

Simplify

x = - \frac{arcsin ( \frac{1}{3} )}{2} + πn

x = - \frac{arcsin ( \frac{1}{3} )}{2} + πn

Solve

2 x = π + arcsin (\frac{1}{3}) + 2 πn : x = \frac{π}{2} + \frac{arcsin ( \frac{1}{3} )}{2} + πn

2 x = π + arcsin (\frac{1}{3}) + 2 πn

Divide both sides by

2

2 x = π + arcsin (\frac{1}{3}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{π}{2} + \frac{arcsin ( \frac{1}{3} )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{π}{2} + \frac{arcsin ( \frac{1}{3} )}{2} + πn

x = \frac{π}{2} + \frac{arcsin ( \frac{1}{3} )}{2} + πn

x = - \frac{arcsin ( \frac{1}{3} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{1}{3} )}{2} + πn

Show solutions in decimal form

x = - \frac{0.33983 \dots}{2} + πn, x = \frac{π}{2} + \frac{0.33983 \dots}{2} + πn

- cos (3 x) = - cos (x)

2 sin (θ) cos (θ) - 2 cos (θ) = 0

4 sinh^{2} (x) = 0

- 2 cos (C) + 5 = 2 cos (C) + 4

2 cos^{2} (θ) + 2 sin (θ) - 12 = 9 sin (θ) - 7

What is the general solution for sin(2x)=-1/3 ?
The general solution for sin(2x)=-1/3 is x=-(0.33983…)/2+pin,x= pi/2+(0.33983…)/2+pin