What is the general solution for 3sin(x)-2sin(x)cos(x)=0 ?

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

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3sin(x)-2sin(x)cos(x)=0

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

x = 2 πn, x = π + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Factor

3 sin (x) - 2 sin (x) cos (x) : - sin (x) (2 cos (x) - 3)

3 sin (x) - 2 sin (x) cos (x)

Factor out common term

- sin (x)

= - sin (x) (- 3 + 2 cos (x))

- sin (x) (2 cos (x) - 3) = 0

Solving each part separately

sin (x) = 0 or 2 cos (x) - 3 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

2 cos (x) - 3 = 0 :

No Solution

2 cos (x) - 3 = 0

Move

3

to the right side

2 cos (x) - 3 = 0

Add

3

to both sides

2 cos (x) - 3 + 3 = 0 + 3

Simplify

2 cos (x) = 3

2 cos (x) = 3

Divide both sides by

2

2 cos (x) = 3

Divide both sides by

2

\frac{2 cos ( x )}{2} = \frac{3}{2}

Simplify

cos (x) = \frac{3}{2}

cos (x) = \frac{3}{2}

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

x = 2 πn, x = π + 2 πn

8 tan (b) + 12 = 3 tan (b) + 7

sin (θ) + 2 = - sin (θ)

cos (x) = π

3 cot (θ) + 1 = 0

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

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