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

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

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cos(x)-1=-cos(x)

cos (x) - 1 = - cos (x)

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

cos (x) - 1 = - cos (x)

Solve by substitution

cos (x) - 1 = - cos (x)

Let:

cos (x) = u

u - 1 = - u

u - 1 = - u : u = \frac{1}{2}

u - 1 = - u

Move

1

to the right side

u - 1 = - u

Add

1

to both sides

u - 1 + 1 = - u + 1

Simplify

u = - u + 1

u = - u + 1

Move

u

to the left side

u = - u + 1

Add

u

to both sides

u + u = - u + 1 + u

Simplify

2 u = 1

2 u = 1

Divide both sides by

2

2 u = 1

Divide both sides by

2

\frac{2 u}{2} = \frac{1}{2}

Simplify

u = \frac{1}{2}

u = \frac{1}{2}

Substitute back

u = cos (x)

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

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

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

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

2 cos^{2} (x) = 3

9 tan (2 x) - 9 cot (x) = 0

sin (2 x) = 2 sin^{2} (x)

sin (x) sec (x) = sin (x)

cos (x) + 2 = 0

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