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

The general solution for sin(2x)=-sqrt(3)sin(x) is x=2pin,x=pi+2pin,x=(5pi)/6+2pin,x=(7pi)/6+2pin

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Popular >

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

Solution

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

Solution

x = 2 πn, x = π + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

- 3 sin (x)

from both sides

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

Rewrite using trig identities

sin (2 x) + sin (x) 3

Use the Double Angle identity:

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

= 2 sin (x) cos (x) + 3 sin (x)

sin (x) 3 + 2 cos (x) sin (x) = 0

Factor

sin (x) 3 + 2 cos (x) sin (x) : sin (x) (3 + 2 cos (x))

sin (x) 3 + 2 cos (x) sin (x)

Factor out common term

sin (x)

= sin (x) (3 + 2 cos (x))

sin (x) (3 + 2 cos (x)) = 0

Solving each part separately

sin (x) = 0 or 3 + 2 cos (x) = 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

3 + 2 cos (x) = 0 : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

3 + 2 cos (x) = 0

Move

3

to the right side

3 + 2 cos (x) = 0

Subtract

3

from both sides

3 + 2 cos (x) - 3 = 0 - 3

Simplify

2 cos (x) = - 3

2 cos (x) = - 3

Divide both sides by

2

2 cos (x) = - 3

Divide both sides by

2

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

Simplify

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

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

General solutions for

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

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

Popular Examples

2cos((x+pi)/6)=1,0<x<2pi

2 cos (\frac{x + π}{6}) = 1, 0 < x < 2 π

tan^2(x)+sec(x)-1=0

tan^{2} (x) + sec (x) - 1 = 0

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

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

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

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

cos(2θ)=(-1)/2

cos (2 θ) = \frac{- 1}{2}

Frequently Asked Questions (FAQ)

What is the general solution for sin(2x)=-sqrt(3)sin(x) ?
The general solution for sin(2x)=-sqrt(3)sin(x) is x=2pin,x=pi+2pin,x=(5pi)/6+2pin,x=(7pi)/6+2pin