What is the general solution for cos(x/2)=sqrt((1-cos(x))/2) ?

The general solution for cos(x/2)=sqrt((1-cos(x))/2) is x= pi/2+4pin,x=4pi-pi/2+4pin

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cos(x/2)=sqrt((1-cos(x))/2)

Solution

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

Solution

x = \frac{π}{2} + 4 πn, x = 4 π - \frac{π}{2} + 4 πn

Degrees

x = 9 0^{\circ} + 72 0^{\circ} n, x = 63 0^{\circ} + 72 0^{\circ} n

Solution steps

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

Subtract

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

from both sides

cos (\frac{x}{2}) - \frac{1 - cos ( x )}{2} = 0

Let:

u = \frac{x}{2}

cos (u) - \frac{1 - cos ( 2 u )}{2} = 0

Rewrite using trig identities

cos (u) - \frac{1 - cos ( 2 u )}{2}

Use the Double Angle identity:

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

= cos (u) - \frac{1 - ( 2 cos ^{2} ( u ) - 1 )}{2}

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

\frac{1 - ( 2 cos ^{2} ( u ) - 1 )}{2}

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

\frac{1 - ( 2 cos ^{2} ( u ) - 1 )}{2}

Expand

1 - (2 cos^{2} (u) - 1) : - 2 cos^{2} (u) + 2

1 - (2 cos^{2} (u) - 1)

- (2 cos^{2} (u) - 1) : - 2 cos^{2} (u) + 1

- (2 cos^{2} (u) - 1)

Distribute parentheses

= - (2 cos^{2} (u)) - (- 1)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 2 cos^{2} (u) + 1

= 1 - 2 cos^{2} (u) + 1

Simplify

1 - 2 cos^{2} (u) + 1 : - 2 cos^{2} (u) + 2

1 - 2 cos^{2} (u) + 1

Group like terms

= - 2 cos^{2} (u) + 1 + 1

Add the numbers:

1 + 1 = 2

= - 2 cos^{2} (u) + 2

= - 2 cos^{2} (u) + 2

= \frac{- 2 cos ^{2} ( u ) + 2}{2}

Factor

- 2 cos^{2} (u) + 2 : 2 (- cos^{2} (u) + 1)

- 2 cos^{2} (u) + 2

Rewrite as

= - 2 cos^{2} (u) + 2 \cdot 1

Factor out common term

2

= 2 (- cos^{2} (u) + 1)

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

Divide the numbers:

\frac{2}{2} = 1

= - cos^{2} (u) + 1

= - cos^{2} (u) + 1

Remove parentheses:

(- a) = - a

= - cos^{2} (u) + 1

= cos (u) - - cos^{2} (u) + 1

cos (u) - 1 - cos^{2} (u) = 0

Solve by substitution

cos (u) - 1 - cos^{2} (u) = 0

Let:

cos (u) = u

u - 1 - u^{2} = 0

u - 1 - u^{2} = 0 : u = \frac{1}{2}

u - 1 - u^{2} = 0

Remove square roots

u - 1 - u^{2} = 0

Subtract

u

from both sides

u - 1 - u^{2} - u = 0 - u

Simplify

- 1 - u^{2} = - u

Square both sides:

1 - u^{2} = u^{2}

u - 1 - u^{2} = 0

(- 1 - u^{2})^{2} = (- u)^{2}

Expand

(- 1 - u^{2})^{2} : 1 - u^{2}

(- 1 - u^{2})^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 1 - u^{2})^{2} = (1 - u^{2})^{2}

= (1 - u^{2})^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= ((1 - u^{2})^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= (1 - u^{2})^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 1 - u^{2}

Expand

(- u)^{2} : u^{2}

(- u)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- u)^{2} = u^{2}

= u^{2}

1 - u^{2} = u^{2}

1 - u^{2} = u^{2}

1 - u^{2} = u^{2}

Solve

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

1 - u^{2} = u^{2}

Move

1

to the right side

1 - u^{2} = u^{2}

Subtract

1

from both sides

1 - u^{2} - 1 = u^{2} - 1

Simplify

- u^{2} = u^{2} - 1

- u^{2} = u^{2} - 1

Move

u^{2}

to the left side

- u^{2} = u^{2} - 1

Subtract

u^{2}

from both sides

- u^{2} - u^{2} = u^{2} - 1 - u^{2}

Simplify

- 2 u^{2} = - 1

- 2 u^{2} = - 1

Divide both sides by

- 2

- 2 u^{2} = - 1

Divide both sides by

- 2

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

Simplify

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

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

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

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

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

Verify Solutions:

u = \frac{1}{2}

True

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

False

Check the solutions by plugging them into

u - 1 - u^{2} = 0

Remove the ones that don't agree with the equation.

Plug in

u = \frac{1}{2} :

True

\frac{1}{2} - 1 - (\frac{1}{2})^{2} = 0

\frac{1}{2} - 1 - (\frac{1}{2})^{2} = 0

\frac{1}{2} - 1 - (\frac{1}{2})^{2}

1 - (\frac{1}{2})^{2} = \frac{1}{2}

1 - (\frac{1}{2})^{2}

(\frac{1}{2})^{2} = \frac{1}{2}

(\frac{1}{2})^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= ((\frac{1}{2})^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= (\frac{1}{2})^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= \frac{1}{2}

= 1 - \frac{1}{2}

Join

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

1 - \frac{1}{2}

Convert element to fraction:

1 = \frac{1 \cdot 2}{2}

= \frac{1 \cdot 2}{2} - \frac{1}{2}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 2 - 1}{2}

1 \cdot 2 - 1 = 1

1 \cdot 2 - 1

Multiply the numbers:

1 \cdot 2 = 2

= 2 - 1

Subtract the numbers:

2 - 1 = 1

= 1

= \frac{1}{2}

= \frac{1}{2}

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

Add similar elements:

\frac{1}{2} - \frac{1}{2} = 0

= 0

0 = 0

True

Plug in

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

False

(- \frac{1}{2}) - 1 - (- \frac{1}{2})^{2} = 0

(- \frac{1}{2}) - 1 - (- \frac{1}{2})^{2} = - 2

(- \frac{1}{2}) - 1 - (- \frac{1}{2})^{2}

Remove parentheses:

(- a) = - a

= - \frac{1}{2} - 1 - (- \frac{1}{2})^{2}

1 - (- \frac{1}{2})^{2} = \frac{1}{2}

1 - (- \frac{1}{2})^{2}

(- \frac{1}{2})^{2} = \frac{1}{2}

(- \frac{1}{2})^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- \frac{1}{2})^{2} = (\frac{1}{2})^{2}

= (\frac{1}{2})^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= ((\frac{1}{2})^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= (\frac{1}{2})^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= \frac{1}{2}

= 1 - \frac{1}{2}

Join

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

1 - \frac{1}{2}

Convert element to fraction:

1 = \frac{1 \cdot 2}{2}

= \frac{1 \cdot 2}{2} - \frac{1}{2}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 2 - 1}{2}

1 \cdot 2 - 1 = 1

1 \cdot 2 - 1

Multiply the numbers:

1 \cdot 2 = 2

= 2 - 1

Subtract the numbers:

2 - 1 = 1

= 1

= \frac{1}{2}

= \frac{1}{2}

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

Add similar elements:

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

= - 2 \frac{1}{2}

Apply exponent rule:

a \frac{b}{a} = a^{2} \frac{b}{a}

2 \frac{1}{2} = 2^{2} \cdot \frac{1}{2}

= - 2^{2} \cdot \frac{1}{2}

Multiply

2^{2} \cdot \frac{1}{2} : 2

2^{2} \cdot \frac{1}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2 ^{2}}{2}

Multiply:

1 \cdot 2^{2} = 2^{2}

= \frac{2 ^{2}}{2}

Cancel the common factor:

2

= 2

= - 2

- 2 = 0

False

The solution is

u = \frac{1}{2}

Substitute back

u = cos (u)

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

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

cos (u) = \frac{1}{2} : u = arccos (\frac{1}{2}) + 2 πn, u = 2 π - arccos (\frac{1}{2}) + 2 πn

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

Apply trig inverse properties

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

General solutions for

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

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

u = arccos (\frac{1}{2}) + 2 πn, u = 2 π - arccos (\frac{1}{2}) + 2 πn

u = arccos (\frac{1}{2}) + 2 πn, u = 2 π - arccos (\frac{1}{2}) + 2 πn

Combine all the solutions

u = arccos (\frac{1}{2}) + 2 πn, u = 2 π - arccos (\frac{1}{2}) + 2 πn

Substitute back

u = \frac{x}{2}

\frac{x}{2} = arccos (\frac{1}{2}) + 2 πn : x = \frac{π}{2} + 4 πn

\frac{x}{2} = arccos (\frac{1}{2}) + 2 πn

Simplify

arccos (\frac{1}{2}) + 2 πn : \frac{π}{4} + 2 πn

arccos (\frac{1}{2}) + 2 πn

Use the following trivial identity:

arccos (\frac{1}{2}) = \frac{π}{4}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= \frac{π}{4} + 2 πn

\frac{x}{2} = \frac{π}{4} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{π}{4} + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot \frac{π}{4} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 \cdot \frac{π}{4} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{π}{4} + 2 \cdot 2 πn : \frac{π}{2} + 4 πn

2 \cdot \frac{π}{4} + 2 \cdot 2 πn

2 \cdot \frac{π}{4} = \frac{π}{2}

2 \cdot \frac{π}{4}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{π 2}{4}

Cancel the common factor:

2

= \frac{π}{2}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{π}{2} + 4 πn

x = \frac{π}{2} + 4 πn

x = \frac{π}{2} + 4 πn

x = \frac{π}{2} + 4 πn

\frac{x}{2} = 2 π - arccos (\frac{1}{2}) + 2 πn : x = 4 π - \frac{π}{2} + 4 πn

\frac{x}{2} = 2 π - arccos (\frac{1}{2}) + 2 πn

Simplify

2 π - arccos (\frac{1}{2}) + 2 πn : 2 π - \frac{π}{4} + 2 πn

2 π - arccos (\frac{1}{2}) + 2 πn

Use the following trivial identity:

arccos (\frac{1}{2}) = \frac{π}{4}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= 2 π - \frac{π}{4} + 2 πn

\frac{x}{2} = 2 π - \frac{π}{4} + 2 πn

Multiply both sides by

2

\frac{x}{2} = 2 π - \frac{π}{4} + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot 2 π - 2 \cdot \frac{π}{4} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 \cdot 2 π - 2 \cdot \frac{π}{4} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot 2 π - 2 \cdot \frac{π}{4} + 2 \cdot 2 πn : 4 π - \frac{π}{2} + 4 πn

2 \cdot 2 π - 2 \cdot \frac{π}{4} + 2 \cdot 2 πn

2 \cdot 2 π = 4 π

2 \cdot 2 π

Multiply the numbers:

2 \cdot 2 = 4

= 4 π

2 \cdot \frac{π}{4} = \frac{π}{2}

2 \cdot \frac{π}{4}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{π 2}{4}

Cancel the common factor:

2

= \frac{π}{2}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= 4 π - \frac{π}{2} + 4 πn

x = 4 π - \frac{π}{2} + 4 πn

x = 4 π - \frac{π}{2} + 4 πn

x = 4 π - \frac{π}{2} + 4 πn

x = \frac{π}{2} + 4 πn, x = 4 π - \frac{π}{2} + 4 πn

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Frequently Asked Questions (FAQ)

What is the general solution for cos(x/2)=sqrt((1-cos(x))/2) ?
The general solution for cos(x/2)=sqrt((1-cos(x))/2) is x= pi/2+4pin,x=4pi-pi/2+4pin