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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

u

to the left side

u = 1 - u

Add

u

to both sides

u + u = 1 - u + 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 (\frac{θ}{2}) = - \frac{1}{2}

sec^{2} (x) = 3 tan (x) + 1

- 3 cos (2 θ) - sin (θ) - 6 = - 8

sin^{2} (x) + 1 = \frac{7}{2} cos^{2} (x)

cos (θ) sin (θ) - 2 cos (θ) = 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