What is the general solution for 3sin(2x)-sin(x)=0 ?

The general solution for 3sin(2x)-sin(x)=0 is x=2pin,x=pi+2pin,x=1.40334…+2pin,x=2pi-1.40334…+2pin

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3sin(2x)-sin(x)=0

Solution

3 sin (2 x) - sin (x) = 0

Solution

x = 2 πn, x = π + 2 πn, x = 1.40334 \dots + 2 πn, x = 2 π - 1.40334 \dots + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 80.40593 \dots^{\circ} + 36 0^{\circ} n, x = 279.59406 \dots^{\circ} + 36 0^{\circ} n

Solution steps

3 sin (2 x) - sin (x) = 0

Rewrite using trig identities

- sin (x) + 3 sin (2 x)

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= - sin (x) + 3 \cdot 2 sin (x) cos (x)

Simplify

= - sin (x) + 6 sin (x) cos (x)

- sin (x) + 6 cos (x) sin (x) = 0

Factor

- sin (x) + 6 cos (x) sin (x) : sin (x) (6 cos (x) - 1)

- sin (x) + 6 cos (x) sin (x)

Factor out common term

sin (x)

= sin (x) (- 1 + 6 cos (x))

sin (x) (6 cos (x) - 1) = 0

Solving each part separately

sin (x) = 0 or 6 cos (x) - 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

6 cos (x) - 1 = 0 : x = arccos (\frac{1}{6}) + 2 πn, x = 2 π - arccos (\frac{1}{6}) + 2 πn

6 cos (x) - 1 = 0

Move

1

to the right side

6 cos (x) - 1 = 0

Add

1

to both sides

6 cos (x) - 1 + 1 = 0 + 1

Simplify

6 cos (x) = 1

6 cos (x) = 1

Divide both sides by

6

6 cos (x) = 1

Divide both sides by

6

\frac{6 cos ( x )}{6} = \frac{1}{6}

Simplify

cos (x) = \frac{1}{6}

cos (x) = \frac{1}{6}

Apply trig inverse properties

cos (x) = \frac{1}{6}

General solutions for

cos (x) = \frac{1}{6}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{1}{6}) + 2 πn, x = 2 π - arccos (\frac{1}{6}) + 2 πn

x = arccos (\frac{1}{6}) + 2 πn, x = 2 π - arccos (\frac{1}{6}) + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arccos (\frac{1}{6}) + 2 πn, x = 2 π - arccos (\frac{1}{6}) + 2 πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 1.40334 \dots + 2 πn, x = 2 π - 1.40334 \dots + 2 πn

Popular Examples

(-6cos(x)-5sin(x))^2+11sin^2(x)=51

(- 6 cos (x) - 5 sin (x))^{2} + 11 sin^{2} (x) = 51

cos(5x)-sin(2x)-cos(x)=0

cos (5 x) - sin (2 x) - cos (x) = 0

cosh(4x)=0

cosh (4 x) = 0

solvefor x,tan(2x)=-(((30+20)/2))/(80)solve for

x, tan (2 x) = - \frac{( \frac{30 + 20}{2} )}{80}

4cos(3θ)=2

4 cos (3 θ) = 2

Frequently Asked Questions (FAQ)

What is the general solution for 3sin(2x)-sin(x)=0 ?
The general solution for 3sin(2x)-sin(x)=0 is x=2pin,x=pi+2pin,x=1.40334…+2pin,x=2pi-1.40334…+2pin