What is the general solution for 3|sin(3θ)|=0 ?

The general solution for 3|sin(3θ)|=0 is θ=(2pin)/3 ,θ= pi/3+(2pin)/3

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3|sin(3θ)|=0

3 ∣ sin (3 θ) ∣ = 0

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

Degrees

θ = 0^{\circ} + 12 0^{\circ} n, θ = 6 0^{\circ} + 12 0^{\circ} n

Solution steps

3 ∣ sin (3 θ) ∣ = 0

Divide both sides by

3

3 ∣ sin (3 θ) ∣ = 0

Divide both sides by

3

\frac{3 ∣ sin ( 3 θ ) ∣}{3} = \frac{0}{3}

Simplify

∣ sin (3 θ) ∣ = 0

∣ sin (3 θ) ∣ = 0

Apply absolute rule: If

∣ u ∣ = 0

then

u = 0

sin (3 θ) = 0

General solutions for

sin (3 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}

3 θ = 0 + 2 πn, 3 θ = π + 2 πn

3 θ = 0 + 2 πn, 3 θ = π + 2 πn

Solve

3 θ = 0 + 2 πn : θ = \frac{2 πn}{3}

3 θ = 0 + 2 πn

0 + 2 πn = 2 πn

3 θ = 2 πn

Divide both sides by

3

3 θ = 2 πn

Divide both sides by

3

\frac{3 θ}{3} = \frac{2 πn}{3}

Simplify

θ = \frac{2 πn}{3}

θ = \frac{2 πn}{3}

Solve

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

3 θ = π + 2 πn

Divide both sides by

3

3 θ = π + 2 πn

Divide both sides by

3

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

Simplify

θ = \frac{π}{3} + \frac{2 πn}{3}

θ = \frac{π}{3} + \frac{2 πn}{3}

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

- 2 cos (x) - 4 sin (2 x) = 0

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

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

2 tan^{2} (x) + 4 sec (x) + sec^{2} (x) = 2

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

What is the general solution for 3|sin(3θ)|=0 ?
The general solution for 3|sin(3θ)|=0 is θ=(2pin)/3 ,θ= pi/3+(2pin)/3