What is the general solution for sin(5x)cos(x)-cos(5x)sin(x)=0 ?

The general solution for sin(5x)cos(x)-cos(5x)sin(x)=0 is x=(pin)/2 ,x= pi/4+(pin)/2

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sin(5x)cos(x)-cos(5x)sin(x)=0

sin (5 x) cos (x) - cos (5 x) sin (x) = 0

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 (5 x) cos (x) - cos (5 x) sin (x) = 0

Rewrite using trig identities

sin (5 x) cos (x) - cos (5 x) sin (x)

Use the Angle Difference identity:

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

= sin (5 x - x)

sin (5 x - x) = 0

General solutions for

sin (5 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}

5 x - x = 0 + 2 πn, 5 x - x = π + 2 πn

5 x - x = 0 + 2 πn, 5 x - x = π + 2 πn

Solve

5 x - x = 0 + 2 πn : x = \frac{πn}{2}

5 x - x = 0 + 2 πn

Add similar elements:

5 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

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

5 x - x = π + 2 πn

Add similar elements:

5 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}

2 sin (3 x + 1) + 1 = 2

2 sin (3 x + 1) + 1 = 1

sin (x) = 6946

sin (2 x) - \frac{3}{2} = 0, - 2 π \leq x \leq 2 π

\frac{sec ( x ) + tan ( x )}{cot ( x ) + cos ( x )} = sec^{2} (x)

What is the general solution for sin(5x)cos(x)-cos(5x)sin(x)=0 ?
The general solution for sin(5x)cos(x)-cos(5x)sin(x)=0 is x=(pin)/2 ,x= pi/4+(pin)/2