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

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

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sin(3θ)cos(θ)-cos(3θ)sin(θ)=(sqrt(3))/2

Solution

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

Solution

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

Degrees

θ = 3 0^{\circ} + 18 0^{\circ} n, θ = 6 0^{\circ} + 18 0^{\circ} n

Solution steps

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

Rewrite using trig identities

sin (3 θ) cos (θ) - cos (3 θ) sin (θ)

Use the Angle Difference identity:

sin (s) cos (t) - cos (s) sin (t) = sin (s - t)

= sin (3 θ - θ)

sin (3 θ - θ) = \frac{3}{2}

General solutions for

sin (3 θ - θ) = \frac{3}{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 - θ = \frac{π}{3} + 2 πn, 3 θ - θ = \frac{2 π}{3} + 2 πn

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

Solve

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

3 θ - θ = \frac{π}{3} + 2 πn

Add similar elements:

3 θ - θ = 2 θ

2 θ = \frac{π}{3} + 2 πn

Divide both sides by

2

2 θ = \frac{π}{3} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

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

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

\frac{\frac{π}{3}}{2} = \frac{π}{6}

\frac{\frac{π}{3}}{2}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{π}{6}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{6} + πn

θ = \frac{π}{6} + πn

θ = \frac{π}{6} + πn

θ = \frac{π}{6} + πn

Solve

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

3 θ - θ = \frac{2 π}{3} + 2 πn

Add similar elements:

3 θ - θ = 2 θ

2 θ = \frac{2 π}{3} + 2 πn

Divide both sides by

2

2 θ = \frac{2 π}{3} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

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

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

\frac{\frac{2 π}{3}}{2} = \frac{π}{3}

\frac{\frac{2 π}{3}}{2}

Apply the fraction rule:

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

= \frac{2 π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{2 π}{6}

Cancel the common factor:

2

= \frac{π}{3}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{3} + πn

θ = \frac{π}{3} + πn

θ = \frac{π}{3} + πn

θ = \frac{π}{3} + πn

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

Graph

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Frequently Asked Questions (FAQ)

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