What is the general solution for sin(6x)cos(x)=-cos(6x)sin(x) ?

The general solution for sin(6x)cos(x)=-cos(6x)sin(x) is x=(2pin)/7 ,x= pi/7+(2pin)/7

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sin(6x)cos(x)=-cos(6x)sin(x)

sin (6 x) cos (x) = - cos (6 x) sin (x)

x = \frac{2 πn}{7}, x = \frac{π}{7} + \frac{2 πn}{7}

Degrees

x = 0^{\circ} + 51.42857 \dots^{\circ} n, x = 25.71428 \dots^{\circ} + 51.42857 \dots^{\circ} n

Solution steps

sin (6 x) cos (x) = - cos (6 x) sin (x)

Subtract

- cos (6 x) sin (x)

from both sides

sin (6 x) cos (x) + cos (6 x) sin (x) = 0

Rewrite using trig identities

sin (6 x) cos (x) + cos (6 x) sin (x)

Use the Angle Sum identity:

sin (s) cos (t) + cos (s) sin (t) = sin (s + t)

= sin (6 x + x)

sin (6 x + x) = 0

General solutions for

sin (6 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}

6 x + x = 0 + 2 πn, 6 x + x = π + 2 πn

6 x + x = 0 + 2 πn, 6 x + x = π + 2 πn

Solve

6 x + x = 0 + 2 πn : x = \frac{2 πn}{7}

6 x + x = 0 + 2 πn

Add similar elements:

6 x + x = 7 x

7 x = 0 + 2 πn

0 + 2 πn = 2 πn

7 x = 2 πn

Divide both sides by

7

7 x = 2 πn

Divide both sides by

7

\frac{7 x}{7} = \frac{2 πn}{7}

Simplify

x = \frac{2 πn}{7}

x = \frac{2 πn}{7}

Solve

6 x + x = π + 2 πn : x = \frac{π}{7} + \frac{2 πn}{7}

6 x + x = π + 2 πn

Add similar elements:

6 x + x = 7 x

7 x = π + 2 πn

Divide both sides by

7

7 x = π + 2 πn

Divide both sides by

7

\frac{7 x}{7} = \frac{π}{7} + \frac{2 πn}{7}

Simplify

x = \frac{π}{7} + \frac{2 πn}{7}

x = \frac{π}{7} + \frac{2 πn}{7}

x = \frac{2 πn}{7}, x = \frac{π}{7} + \frac{2 πn}{7}

x, cos (x y) = 1 + sin (y)

4 sin (x) - 1 = 2 sin (x) + 1

sin (3 x) = \frac{3}{4}

sin (x + \frac{π}{3}) + cos (x + \frac{π}{6}) = \frac{3}{3}

4 sin^{2} (x) cot (x) - 3 cot (x) = 0

What is the general solution for sin(6x)cos(x)=-cos(6x)sin(x) ?
The general solution for sin(6x)cos(x)=-cos(6x)sin(x) is x=(2pin)/7 ,x= pi/7+(2pin)/7