What is the general solution for sin(3x)cos(x)-cos(3x)sin(x)=0 ?

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

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sin(3x)cos(x)-cos(3x)sin(x)=0

sin (3 x) cos (x) - cos (3 x) sin (x) = 0

x = πn, x = \frac{π}{2} + πn

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 9 0^{\circ} + 18 0^{\circ} n

Solution steps

sin (3 x) cos (x) - cos (3 x) sin (x) = 0

Rewrite using trig identities

sin (3 x) cos (x) - cos (3 x) sin (x)

Use the Angle Difference identity:

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

= sin (3 x - x)

sin (3 x - x) = 0

General solutions for

sin (3 x - 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}

3 x - x = 0 + 2 πn, 3 x - x = π + 2 πn

3 x - x = 0 + 2 πn, 3 x - x = π + 2 πn

Solve

3 x - x = 0 + 2 πn : x = πn

3 x - x = 0 + 2 πn

Add similar elements:

3 x - x = 2 x

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

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

3 x - x = π + 2 πn

Add similar elements:

3 x - x = 2 x

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

4 sin (2 x) = - 2

x, sin^{2} (x) = 0

cos (t) = - \frac{3}{4}

sin (2 x) - 2 cos (x) = 2 sin (x) - 1

cos^{2} (θ) - \frac{3}{2} cos (2 θ) = 0

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