What is the general solution for sin(u)-2sin(u)cos(u)=0 ?

The general solution for sin(u)-2sin(u)cos(u)=0 is u=2pin,u=pi+2pin,u= pi/3+2pin,u=(5pi)/3+2pin

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

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

u = 2 πn, u = π + 2 πn, u = \frac{π}{3} + 2 πn, u = \frac{5 π}{3} + 2 πn

Degrees

u = 0^{\circ} + 36 0^{\circ} n, u = 18 0^{\circ} + 36 0^{\circ} n, u = 6 0^{\circ} + 36 0^{\circ} n, u = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Factor

sin (u) - 2 sin (u) cos (u) : - sin (u) (2 cos (u) - 1)

sin (u) - 2 sin (u) cos (u)

Factor out common term

- sin (u)

= - sin (u) (- 1 + 2 cos (u))

- sin (u) (2 cos (u) - 1) = 0

Solving each part separately

sin (u) = 0 or 2 cos (u) - 1 = 0

sin (u) = 0 : u = 2 πn, u = π + 2 πn

sin (u) = 0

General solutions for

sin (u) = 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}

u = 0 + 2 πn, u = π + 2 πn

u = 0 + 2 πn, u = π + 2 πn

Solve

u = 0 + 2 πn : u = 2 πn

u = 0 + 2 πn

0 + 2 πn = 2 πn

u = 2 πn

u = 2 πn, u = π + 2 πn

2 cos (u) - 1 = 0 : u = \frac{π}{3} + 2 πn, u = \frac{5 π}{3} + 2 πn

2 cos (u) - 1 = 0

Move

1

to the right side

2 cos (u) - 1 = 0

Add

1

to both sides

2 cos (u) - 1 + 1 = 0 + 1

Simplify

2 cos (u) = 1

2 cos (u) = 1

Divide both sides by

2

2 cos (u) = 1

Divide both sides by

2

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

Simplify

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

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

General solutions for

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

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

u = \frac{π}{3} + 2 πn, u = \frac{5 π}{3} + 2 πn

u = \frac{π}{3} + 2 πn, u = \frac{5 π}{3} + 2 πn

Combine all the solutions

u = 2 πn, u = π + 2 πn, u = \frac{π}{3} + 2 πn, u = \frac{5 π}{3} + 2 πn

20 = 27 sin (x) - 1.5 cos (x)

- 3 sin (θ) = 0

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

- 2 tan (θ) - 4 - tan^{2} (θ) = - 3

cos^{2} (x) = 3 - 5 cos (x)

What is the general solution for sin(u)-2sin(u)cos(u)=0 ?
The general solution for sin(u)-2sin(u)cos(u)=0 is u=2pin,u=pi+2pin,u= pi/3+2pin,u=(5pi)/3+2pin