What is the general solution for sin(3x)+sin(x)=0,0<= x<= 2pi ?

Q: What is the general solution for sin(3x)+sin(x)=0,0<= x<= 2pi ?

The general solution for sin(3x)+sin(x)=0,0<= x<= 2pi is x= pi/2 ,x=(3pi)/2 ,x=0,x=pi,x=2pi

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sin(3x)+sin(x)=0,0<= x<= 2pi

Solution

sin (3 x) + sin (x) = 0, 0 \leq x \leq 2 π

Solution

x = \frac{π}{2}, x = \frac{3 π}{2}, x = 0, x = π, x = 2 π

Degrees

x = 9 0^{\circ}, x = 27 0^{\circ}, x = 0^{\circ}, x = 18 0^{\circ}, x = 36 0^{\circ}

Solution steps

sin (3 x) + sin (x) = 0, 0 \leq x \leq 2 π

Rewrite using trig identities

sin (3 x) + sin (x)

Use the Sum to Product identity:

sin (s) + sin (t) = 2 sin (\frac{s + t}{2}) cos (\frac{s - t}{2})

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

Simplify

2 sin (\frac{3 x + x}{2}) cos (\frac{3 x - x}{2}) : 2 sin (2 x) cos (x)

2 sin (\frac{3 x + x}{2}) cos (\frac{3 x - x}{2})

\frac{3 x + x}{2} = 2 x

\frac{3 x + x}{2}

Add similar elements:

3 x + x = 4 x

= \frac{4 x}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 x

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

\frac{3 x - x}{2} = x

\frac{3 x - x}{2}

Add similar elements:

3 x - x = 2 x

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

= 2 sin (2 x) cos (x)

= 2 sin (2 x) cos (x)

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

Solving each part separately

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

cos (x) = 0, 0 \leq x \leq 2 π : x = \frac{π}{2}, x = \frac{3 π}{2}

cos (x) = 0, 0 \leq x \leq 2 π

General solutions for

cos (x) = 0

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{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Solutions for the range

0 \leq x \leq 2 π

x = \frac{π}{2}, x = \frac{3 π}{2}

sin (2 x) = 0, 0 \leq x \leq 2 π : x = 0, x = \frac{π}{2}, x = π, x = \frac{3 π}{2}, x = 2 π

sin (2 x) = 0, 0 \leq x \leq 2 π

General solutions for

sin (2 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}

2 x = 0 + 2 πn, 2 x = π + 2 πn

2 x = 0 + 2 πn, 2 x = π + 2 πn

Solve

2 x = 0 + 2 πn : x = πn

2 x = 0 + 2 πn

0 + 2 πn = 2 πn

2 x = 2 πn

Divide both sides by

2

2 x = 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{2 πn}{2}

Simplify

x = πn

x = πn

Solve

2 x = π + 2 πn : x = \frac{π}{2} + πn

2 x = π + 2 πn

Divide both sides by

2

2 x = π + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{π}{2} + \frac{2 πn}{2}

Simplify

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

x = πn, x = \frac{π}{2} + πn

Solutions for the range

0 \leq x \leq 2 π

x = 0, x = \frac{π}{2}, x = π, x = \frac{3 π}{2}, x = 2 π

Combine all the solutions

x = \frac{π}{2}, x = \frac{3 π}{2}, x = 0, x = π, x = 2 π

Popular Examples

4sin(θ)+3=1

4 sin (θ) + 3 = 1

25cot^2(θ)-1=0

25 cot^{2} (θ) - 1 = 0

cos(a)= 5/13

cos (a) = \frac{5}{13}

sec(2x)-2=0

sec (2 x) - 2 = 0

2sin^2(θ)+sin(θ)=1

2 sin^{2} (θ) + sin (θ) = 1

Frequently Asked Questions (FAQ)

What is the general solution for sin(3x)+sin(x)=0,0<= x<= 2pi ?
The general solution for sin(3x)+sin(x)=0,0<= x<= 2pi is x= pi/2 ,x=(3pi)/2 ,x=0,x=pi,x=2pi