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

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

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sin(4t)cos(t)=sin(t)cos(4t)

sin (4 t) cos (t) = sin (t) cos (4 t)

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

Degrees

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

Solution steps

sin (4 t) cos (t) = sin (t) cos (4 t)

Subtract

sin (t) cos (4 t)

from both sides

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

Rewrite using trig identities

sin (4 t) cos (t) - sin (t) cos (4 t)

Use the Angle Difference identity:

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

= sin (4 t - t)

sin (4 t - t) = 0

General solutions for

sin (4 t - t) = 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 t - t = 0 + 2 πn, 4 t - t = π + 2 πn

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

Solve

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

4 t - t = 0 + 2 πn

Add similar elements:

4 t - t = 3 t

3 t = 0 + 2 πn

0 + 2 πn = 2 πn

3 t = 2 πn

Divide both sides by

3

3 t = 2 πn

Divide both sides by

3

\frac{3 t}{3} = \frac{2 πn}{3}

Simplify

t = \frac{2 πn}{3}

t = \frac{2 πn}{3}

Solve

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

4 t - t = π + 2 πn

Add similar elements:

4 t - t = 3 t

3 t = π + 2 πn

Divide both sides by

3

3 t = π + 2 πn

Divide both sides by

3

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

Simplify

t = \frac{π}{3} + \frac{2 πn}{3}

t = \frac{π}{3} + \frac{2 πn}{3}

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

3 tan (3 x + \frac{π}{2}) + 1 = 0

- 2 sin (x) - 2 cos (x) = 0

7 sin (b) = 5 sin (7 0^{\circ})

5 sin (x) + 2 cos^{2} (x) = 4

cos (\frac{1}{2} x) = - 1, y = 2

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