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

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

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1-cos(x)=cos(x)

1 - cos (x) = 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

1 - cos (x) = cos (x)

Solve by substitution

1 - cos (x) = cos (x)

Let:

cos (x) = u

1 - u = u

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

1 - u = u

Move

1

to the right side

1 - u = u

Subtract

1

from both sides

1 - u - 1 = u - 1

Simplify

- u = u - 1

- u = u - 1

Move

u

to the left side

- u = u - 1

Subtract

u

from 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

sin^{2} (x) - 1 = 0, 0 \leq x \leq 2 π

2 cos^{3} (θ) = cos (θ)

t, x = arccos (\frac{1}{1 + t ^{2}})

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

(1 - sin (x)) (1 + cos (x)) = cos^{2} (x)

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