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

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

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

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

x = \frac{πn}{2}, x = \frac{π}{4} + \frac{πn}{2}

Degrees

x = 0^{\circ} + 9 0^{\circ} n, x = 4 5^{\circ} + 9 0^{\circ} n

Solution steps

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

Subtract

- cos (3 x) sin (x)

from both sides

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 Sum 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 = \frac{πn}{2}

3 x + x = 0 + 2 πn

Add similar elements:

3 x + x = 4 x

4 x = 0 + 2 πn

0 + 2 πn = 2 πn

4 x = 2 πn

Divide both sides by

4

4 x = 2 πn

Divide both sides by

4

\frac{4 x}{4} = \frac{2 πn}{4}

Simplify

x = \frac{πn}{2}

x = \frac{πn}{2}

Solve

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

3 x + x = π + 2 πn

Add similar elements:

3 x + x = 4 x

4 x = π + 2 πn

Divide both sides by

4

4 x = π + 2 πn

Divide both sides by

4

\frac{4 x}{4} = \frac{π}{4} + \frac{2 πn}{4}

Simplify

x = \frac{π}{4} + \frac{πn}{2}

x = \frac{π}{4} + \frac{πn}{2}

x = \frac{πn}{2}, x = \frac{π}{4} + \frac{πn}{2}

6 sin^{2} (x) + sin (x) - 1 = 0

sin (2 x) - 2 cos^{2} (x) = 0

sin^{2} (x) - 7 cos (x) + 7 = 0

sin (x) - 3 = - sin (x)

6 sin (θ) - 6 = 0

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