What is the general solution for sin(3θ)=(sqrt(2))/2 ?

The general solution for sin(3θ)=(sqrt(2))/2 is θ= pi/(12)+(2pin)/3 ,θ= pi/4+(2pin)/3

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sin(3θ)=(sqrt(2))/2

Solution

sin (3 θ) = \frac{2}{2}

Solution

θ = \frac{π}{12} + \frac{2 πn}{3}, θ = \frac{π}{4} + \frac{2 πn}{3}

Degrees

θ = 1 5^{\circ} + 12 0^{\circ} n, θ = 4 5^{\circ} + 12 0^{\circ} n

Solution steps

sin (3 θ) = \frac{2}{2}

General solutions for

sin (3 θ) = \frac{2}{2}

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 = \frac{π}{4} + 2 πn, 3 θ = \frac{3 π}{4} + 2 πn

3 θ = \frac{π}{4} + 2 πn, 3 θ = \frac{3 π}{4} + 2 πn

Solve

3 θ = \frac{π}{4} + 2 πn : θ = \frac{π}{12} + \frac{2 πn}{3}

3 θ = \frac{π}{4} + 2 πn

Divide both sides by

3

3 θ = \frac{π}{4} + 2 πn

Divide both sides by

3

\frac{3 θ}{3} = \frac{\frac{π}{4}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 θ}{3} = \frac{\frac{π}{4}}{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}{4}}{3} + \frac{2 πn}{3} : \frac{π}{12} + \frac{2 πn}{3}

\frac{\frac{π}{4}}{3} + \frac{2 πn}{3}

\frac{\frac{π}{4}}{3} = \frac{π}{12}

\frac{\frac{π}{4}}{3}

Apply the fraction rule:

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

= \frac{π}{4 \cdot 3}

Multiply the numbers:

4 \cdot 3 = 12

= \frac{π}{12}

= \frac{π}{12} + \frac{2 πn}{3}

θ = \frac{π}{12} + \frac{2 πn}{3}

θ = \frac{π}{12} + \frac{2 πn}{3}

θ = \frac{π}{12} + \frac{2 πn}{3}

Solve

3 θ = \frac{3 π}{4} + 2 πn : θ = \frac{π}{4} + \frac{2 πn}{3}

3 θ = \frac{3 π}{4} + 2 πn

Divide both sides by

3

3 θ = \frac{3 π}{4} + 2 πn

Divide both sides by

3

\frac{3 θ}{3} = \frac{\frac{3 π}{4}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 θ}{3} = \frac{\frac{3 π}{4}}{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}{4}}{3} + \frac{2 πn}{3} : \frac{π}{4} + \frac{2 πn}{3}

\frac{\frac{3 π}{4}}{3} + \frac{2 πn}{3}

\frac{\frac{3 π}{4}}{3} = \frac{π}{4}

\frac{\frac{3 π}{4}}{3}

Apply the fraction rule:

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

= \frac{3 π}{4 \cdot 3}

Multiply the numbers:

4 \cdot 3 = 12

= \frac{3 π}{12}

Cancel the common factor:

3

= \frac{π}{4}

= \frac{π}{4} + \frac{2 πn}{3}

θ = \frac{π}{4} + \frac{2 πn}{3}

θ = \frac{π}{4} + \frac{2 πn}{3}

θ = \frac{π}{4} + \frac{2 πn}{3}

θ = \frac{π}{12} + \frac{2 πn}{3}, θ = \frac{π}{4} + \frac{2 πn}{3}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(3θ)=(sqrt(2))/2 ?
The general solution for sin(3θ)=(sqrt(2))/2 is θ= pi/(12)+(2pin)/3 ,θ= pi/4+(2pin)/3