What is the general solution for sin(4θ)=sin(2θ) ?

The general solution for sin(4θ)=sin(2θ) is θ= pi/6+(2pin)/3 ,θ= pi/2+(2pin)/3 ,θ=2pin,θ=pi+2pin

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Popular Trigonometry >

sin(4θ)=sin(2θ)

Solution

sin (4 θ) = sin (2 θ)

Solution

θ = \frac{π}{6} + \frac{2 πn}{3}, θ = \frac{π}{2} + \frac{2 πn}{3}, θ = 2 πn, θ = π + 2 πn

Degrees

θ = 3 0^{\circ} + 12 0^{\circ} n, θ = 9 0^{\circ} + 12 0^{\circ} n, θ = 0^{\circ} + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

sin (4 θ) = sin (2 θ)

Subtract

sin (2 θ)

from both sides

sin (4 θ) - sin (2 θ) = 0

Rewrite using trig identities

- sin (2 θ) + sin (4 θ)

Use the Sum to Product identity:

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

= 2 sin (\frac{4 θ - 2 θ}{2}) cos (\frac{4 θ + 2 θ}{2})

Simplify

2 sin (\frac{4 θ - 2 θ}{2}) cos (\frac{4 θ + 2 θ}{2}) : 2 sin (θ) cos (3 θ)

2 sin (\frac{4 θ - 2 θ}{2}) cos (\frac{4 θ + 2 θ}{2})

\frac{4 θ - 2 θ}{2} = θ

\frac{4 θ - 2 θ}{2}

Add similar elements:

4 θ - 2 θ = 2 θ

= \frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

= 2 sin (θ) cos (\frac{4 θ + 2 θ}{2})

\frac{4 θ + 2 θ}{2} = 3 θ

\frac{4 θ + 2 θ}{2}

Add similar elements:

4 θ + 2 θ = 6 θ

= \frac{6 θ}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 θ

= 2 sin (θ) cos (3 θ)

= 2 sin (θ) cos (3 θ)

2 cos (3 θ) sin (θ) = 0

Solving each part separately

cos (3 θ) = 0 or sin (θ) = 0

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

cos (3 θ) = 0

General solutions for

cos (3 θ) = 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}

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

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

Solve

3 θ = \frac{π}{2} + 2 πn : θ = \frac{π}{6} + \frac{2 πn}{3}

3 θ = \frac{π}{2} + 2 πn

Divide both sides by

3

3 θ = \frac{π}{2} + 2 πn

Divide both sides by

3

\frac{3 θ}{3} = \frac{\frac{π}{2}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 θ}{3} = \frac{\frac{π}{2}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 θ}{3} : θ

\frac{3 θ}{3}

Divide the numbers:

\frac{3}{3} = 1

= θ

Simplify

\frac{\frac{π}{2}}{3} + \frac{2 πn}{3} : \frac{π}{6} + \frac{2 πn}{3}

\frac{\frac{π}{2}}{3} + \frac{2 πn}{3}

\frac{\frac{π}{2}}{3} = \frac{π}{6}

\frac{\frac{π}{2}}{3}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{π}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{π}{6}

= \frac{π}{6} + \frac{2 πn}{3}

θ = \frac{π}{6} + \frac{2 πn}{3}

θ = \frac{π}{6} + \frac{2 πn}{3}

θ = \frac{π}{6} + \frac{2 πn}{3}

Solve

3 θ = \frac{3 π}{2} + 2 πn : θ = \frac{π}{2} + \frac{2 πn}{3}

3 θ = \frac{3 π}{2} + 2 πn

Divide both sides by

3

3 θ = \frac{3 π}{2} + 2 πn

Divide both sides by

3

\frac{3 θ}{3} = \frac{\frac{3 π}{2}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 θ}{3} = \frac{\frac{3 π}{2}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 θ}{3} : θ

\frac{3 θ}{3}

Divide the numbers:

\frac{3}{3} = 1

= θ

Simplify

\frac{\frac{3 π}{2}}{3} + \frac{2 πn}{3} : \frac{π}{2} + \frac{2 πn}{3}

\frac{\frac{3 π}{2}}{3} + \frac{2 πn}{3}

\frac{\frac{3 π}{2}}{3} = \frac{π}{2}

\frac{\frac{3 π}{2}}{3}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{3 π}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{3 π}{6}

Cancel the common factor:

3

= \frac{π}{2}

= \frac{π}{2} + \frac{2 πn}{3}

θ = \frac{π}{2} + \frac{2 πn}{3}

θ = \frac{π}{2} + \frac{2 πn}{3}

θ = \frac{π}{2} + \frac{2 πn}{3}

θ = \frac{π}{6} + \frac{2 πn}{3}, θ = \frac{π}{2} + \frac{2 πn}{3}

sin (θ) = 0 : θ = 2 πn, θ = π + 2 πn

sin (θ) = 0

General solutions for

sin (θ) = 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}

θ = 0 + 2 πn, θ = π + 2 πn

θ = 0 + 2 πn, θ = π + 2 πn

Solve

θ = 0 + 2 πn : θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn, θ = π + 2 πn

Combine all the solutions

θ = \frac{π}{6} + \frac{2 πn}{3}, θ = \frac{π}{2} + \frac{2 πn}{3}, θ = 2 πn, θ = π + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(4θ)=sin(2θ) ?
The general solution for sin(4θ)=sin(2θ) is θ= pi/6+(2pin)/3 ,θ= pi/2+(2pin)/3 ,θ=2pin,θ=pi+2pin