What is the general solution for (1-2cos(x))(1+sin(x))=0 ?

The general solution for (1-2cos(x))(1+sin(x))=0 is x= pi/3+2pin,x=(5pi)/3+2pin,x=(3pi)/2+2pin

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

Solution

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

Solution

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

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Solving each part separately

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

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

1 - 2 cos (x) = 0

Move

1

to the right side

1 - 2 cos (x) = 0

Subtract

1

from both sides

1 - 2 cos (x) - 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

1 + sin (x) = 0 : x = \frac{3 π}{2} + 2 πn

1 + sin (x) = 0

Move

1

to the right side

1 + sin (x) = 0

Subtract

1

from both sides

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

Simplify

sin (x) = - 1

sin (x) = - 1

General solutions for

sin (x) = - 1

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{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

Combine all the solutions

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

Popular Examples

3cos(2x)+7sin(x)-5=0

3 cos (2 x) + 7 sin (x) - 5 = 0

cos^2(θ)= 1/3

cos^{2} (θ) = \frac{1}{3}

solvefor x,sin(x/5)=-(sqrt(3))/2 solve for

x, sin (\frac{x}{5}) = - \frac{3}{2}

sin(B)= 3/10

sin (B) = \frac{3}{10}

sqrt(3)=2sqrt(3)-3cot(x)

3 = 23 - 3 cot (x)

Frequently Asked Questions (FAQ)

What is the general solution for (1-2cos(x))(1+sin(x))=0 ?
The general solution for (1-2cos(x))(1+sin(x))=0 is x= pi/3+2pin,x=(5pi)/3+2pin,x=(3pi)/2+2pin