What is the general solution for 3+sin(2x)=1-6sin(2x) ?

The general solution for 3+sin(2x)=1-6sin(2x) is x=-(0.28975…)/2+pin,x= pi/2+(0.28975…)/2+pin

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3+sin(2x)=1-6sin(2x)

Solution

3 + sin (2 x) = 1 - 6 sin (2 x)

Solution

x = - \frac{0.28975 \dots}{2} + πn, x = \frac{π}{2} + \frac{0.28975 \dots}{2} + πn

Degrees

x = - 8.30077 \dots^{\circ} + 18 0^{\circ} n, x = 98.30077 \dots^{\circ} + 18 0^{\circ} n

Solution steps

3 + sin (2 x) = 1 - 6 sin (2 x)

Solve by substitution

3 + sin (2 x) = 1 - 6 sin (2 x)

Let:

sin (2 x) = u

3 + u = 1 - 6 u

3 + u = 1 - 6 u : u = - \frac{2}{7}

3 + u = 1 - 6 u

Move

3

to the right side

3 + u = 1 - 6 u

Subtract

3

from both sides

3 + u - 3 = 1 - 6 u - 3

Simplify

u = - 6 u - 2

u = - 6 u - 2

Move

6 u

to the left side

u = - 6 u - 2

Add

6 u

to both sides

u + 6 u = - 6 u - 2 + 6 u

Simplify

7 u = - 2

7 u = - 2

Divide both sides by

7

7 u = - 2

Divide both sides by

7

\frac{7 u}{7} = \frac{- 2}{7}

Simplify

u = - \frac{2}{7}

u = - \frac{2}{7}

Substitute back

u = sin (2 x)

sin (2 x) = - \frac{2}{7}

sin (2 x) = - \frac{2}{7}

sin (2 x) = - \frac{2}{7} : x = - \frac{arcsin ( \frac{2}{7} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{2}{7} )}{2} + πn

sin (2 x) = - \frac{2}{7}

Apply trig inverse properties

sin (2 x) = - \frac{2}{7}

General solutions for

sin (2 x) = - \frac{2}{7}

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

2 x = arcsin (- \frac{2}{7}) + 2 πn, 2 x = π + arcsin (\frac{2}{7}) + 2 πn

2 x = arcsin (- \frac{2}{7}) + 2 πn, 2 x = π + arcsin (\frac{2}{7}) + 2 πn

Solve

2 x = arcsin (- \frac{2}{7}) + 2 πn : x = - \frac{arcsin ( \frac{2}{7} )}{2} + πn

2 x = arcsin (- \frac{2}{7}) + 2 πn

Simplify

arcsin (- \frac{2}{7}) + 2 πn : - arcsin (\frac{2}{7}) + 2 πn

arcsin (- \frac{2}{7}) + 2 πn

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{2}{7}) = - arcsin (\frac{2}{7})

= - arcsin (\frac{2}{7}) + 2 πn

2 x = - arcsin (\frac{2}{7}) + 2 πn

Divide both sides by

2

2 x = - arcsin (\frac{2}{7}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = - \frac{arcsin ( \frac{2}{7} )}{2} + \frac{2 πn}{2}

Simplify

x = - \frac{arcsin ( \frac{2}{7} )}{2} + πn

x = - \frac{arcsin ( \frac{2}{7} )}{2} + πn

Solve

2 x = π + arcsin (\frac{2}{7}) + 2 πn : x = \frac{π}{2} + \frac{arcsin ( \frac{2}{7} )}{2} + πn

2 x = π + arcsin (\frac{2}{7}) + 2 πn

Divide both sides by

2

2 x = π + arcsin (\frac{2}{7}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{π}{2} + \frac{arcsin ( \frac{2}{7} )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{π}{2} + \frac{arcsin ( \frac{2}{7} )}{2} + πn

x = \frac{π}{2} + \frac{arcsin ( \frac{2}{7} )}{2} + πn

x = - \frac{arcsin ( \frac{2}{7} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{2}{7} )}{2} + πn

Combine all the solutions

x = - \frac{arcsin ( \frac{2}{7} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{2}{7} )}{2} + πn

Show solutions in decimal form

x = - \frac{0.28975 \dots}{2} + πn, x = \frac{π}{2} + \frac{0.28975 \dots}{2} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 3+sin(2x)=1-6sin(2x) ?
The general solution for 3+sin(2x)=1-6sin(2x) is x=-(0.28975…)/2+pin,x= pi/2+(0.28975…)/2+pin