What is the general solution for 2sin(x)cos(x)-2sin(x)+cos(x)=1 ?

The general solution for 2sin(x)cos(x)-2sin(x)+cos(x)=1 is x=2pin,x=(7pi)/6+2pin,x=(11pi)/6+2pin

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Popular >

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

Solution

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

Solution

x = 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

1

from both sides

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

Factor

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

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

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

Factor out

2 sin (x)

from

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

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

Factor out common term

2 sin (x)

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

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

Factor out common term

cos (x) - 1

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

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

Solving each part separately

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

cos (x) - 1 = 0 : x = 2 πn

cos (x) - 1 = 0

Move

1

to the right side

cos (x) - 1 = 0

Add

1

to both sides

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

Simplify

cos (x) = 1

cos (x) = 1

General solutions for

cos (x) = 1

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 = 0 + 2 πn

x = 0 + 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

2 sin (x) + 1 = 0 : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

2 sin (x) + 1 = 0

Move

1

to the right side

2 sin (x) + 1 = 0

Subtract

1

from both sides

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

Simplify

2 sin (x) = - 1

2 sin (x) = - 1

Divide both sides by

2

2 sin (x) = - 1

Divide both sides by

2

\frac{2 sin ( x )}{2} = \frac{- 1}{2}

Simplify

sin (x) = - \frac{1}{2}

sin (x) = - \frac{1}{2}

General solutions for

sin (x) = - \frac{1}{2}

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 = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

x = 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Popular Examples

8cos(4x)=0

8 cos (4 x) = 0

cos(3x)+sin(5x)=0

cos (3 x) + sin (5 x) = 0

sin^2(o)+3sin(o)-4=0

sin^{2} (o) + 3 sin (o) - 4 = 0

tan(θ)=3.8

tan (θ) = 3.8

solvefor k,f=x^{2/3}+(10-x^2)^{0.5}sin(kpix)solve for

k, f = x^{\frac{2}{3}} + (10 - x^{2})^{0.5} sin (kπ x)

Frequently Asked Questions (FAQ)

What is the general solution for 2sin(x)cos(x)-2sin(x)+cos(x)=1 ?
The general solution for 2sin(x)cos(x)-2sin(x)+cos(x)=1 is x=2pin,x=(7pi)/6+2pin,x=(11pi)/6+2pin