What is the general solution for sin(4x)cos(x)=cos(4x)sin(x) ?

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

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sin(4x)cos(x)=cos(4x)sin(x)

sin (4 x) cos (x) = cos (4 x) sin (x)

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

Degrees

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

Solution steps

sin (4 x) cos (x) = cos (4 x) sin (x)

Subtract

cos (4 x) sin (x)

from both sides

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

Rewrite using trig identities

sin (4 x) cos (x) - cos (4 x) sin (x)

Use the Angle Difference identity:

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

= sin (4 x - x)

sin (4 x - x) = 0

General solutions for

sin (4 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}

4 x - x = 0 + 2 πn, 4 x - x = π + 2 πn

4 x - x = 0 + 2 πn, 4 x - x = π + 2 πn

Solve

4 x - x = 0 + 2 πn : x = \frac{2 πn}{3}

4 x - x = 0 + 2 πn

Add similar elements:

4 x - x = 3 x

3 x = 0 + 2 πn

0 + 2 πn = 2 πn

3 x = 2 πn

Divide both sides by

3

3 x = 2 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{2 πn}{3}

Simplify

x = \frac{2 πn}{3}

x = \frac{2 πn}{3}

Solve

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

4 x - x = π + 2 πn

Add similar elements:

4 x - x = 3 x

3 x = π + 2 πn

Divide both sides by

3

3 x = π + 2 πn

Divide both sides by

3

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

Simplify

x = \frac{π}{3} + \frac{2 πn}{3}

x = \frac{π}{3} + \frac{2 πn}{3}

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

cos (θ) - cos (θ) = 0

2 sin^{2} (x) = \frac{csc ^{2} ( x )}{2}

4 sin^{2} (x) + 4 sin (x) = 1

5 cos (t) = 0

2 cos (\frac{x}{2}) = 1

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