What is the general solution for 2sin(x)cos(x)-4cos(x)-sin(x)+2=0 ?

The general solution for 2sin(x)cos(x)-4cos(x)-sin(x)+2=0 is x= pi/3+2pin,x=(5pi)/3+2pin

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Popular >

2sin(x)cos(x)-4cos(x)-sin(x)+2=0

Solution

2 sin (x) cos (x) - 4 cos (x) - sin (x) + 2 = 0

Solution

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

Degrees

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

Solution steps

2 sin (x) cos (x) - 4 cos (x) - sin (x) + 2 = 0

Factor

2 sin (x) cos (x) - 4 cos (x) - sin (x) + 2 : (sin (x) - 2) (2 cos (x) - 1)

2 sin (x) cos (x) - 4 cos (x) - sin (x) + 2

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

Factor out

- 1

from

- sin (x) + 2 : - (sin (x) - 2)

- sin (x) + 2

Factor out common term

- 1

= - (sin (x) - 2)

Factor out

2 cos (x)

from

2 sin (x) cos (x) - 4 cos (x) : 2 cos (x) (sin (x) - 2)

2 sin (x) cos (x) - 4 cos (x)

Rewrite

- 4

2 \cdot 2

= 2 sin (x) cos (x) + 2 \cdot 2 cos (x)

Factor out common term

2 cos (x)

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

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

Factor out common term

sin (x) - 2

= (sin (x) - 2) (2 cos (x) - 1)

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

Solving each part separately

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

sin (x) - 2 = 0 :

No Solution

sin (x) - 2 = 0

Move

2

to the right side

sin (x) - 2 = 0

Add

2

to both sides

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

Simplify

sin (x) = 2

sin (x) = 2

- 1 \leq sin (x) \leq 1

No Solution

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

2 cos (x) - 1 = 0

Move

1

to the right side

2 cos (x) - 1 = 0

Add

1

to both sides

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

Simplify

2 cos (x) = 1

2 cos (x) = 1

Divide both sides by

2

2 cos (x) = 1

Divide both sides by

2

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

Simplify

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

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

General solutions for

cos (x) = \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}

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

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

Combine all the solutions

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

Popular Examples

(sin(pi/2))/5 =(sin(x))/4

\frac{sin ( \frac{π}{2} )}{5} = \frac{sin ( x )}{4}

cos(3x-1)= 1/2

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

tan^2(x)=sec(x)+1

tan^{2} (x) = sec (x) + 1

cos(u)= 2/5 ,cos(2u)

cos (u) = \frac{2}{5}, cos (2 u)

sin(b)=(sqrt(2))/2

sin (b) = \frac{2}{2}

Frequently Asked Questions (FAQ)

What is the general solution for 2sin(x)cos(x)-4cos(x)-sin(x)+2=0 ?
The general solution for 2sin(x)cos(x)-4cos(x)-sin(x)+2=0 is x= pi/3+2pin,x=(5pi)/3+2pin