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

The general solution for sin^2(x/2)=cos(x)-1 is x=2pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

sin^2(x/2)=cos(x)-1

Solution

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

Solution

x = 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

cos (x) - 1

from both sides

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

Rewrite using trig identities

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

Use the following identity:

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

Use the Double Angle identity

cos (2 θ) = 1 - 2 sin^{2} (θ)

Switch sides

2 sin^{2} (θ) - 1 = - cos (2 θ)

Add

1

to both sides

2 sin^{2} (θ) = 1 - cos (2 θ)

Divide both sides by

2

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

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

Simplify

1 - cos (x) + \frac{1 - cos ( 2 \cdot \frac{x}{2} )}{2} : \frac{1 - 3 cos ( x )}{2} + 1

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

Multiply

2 \cdot \frac{x}{2} : x

2 \cdot \frac{x}{2}

Multiply fractions:

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

= \frac{x \cdot 2}{2}

Cancel the common factor:

2

= x

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

Combine the fractions

\frac{- cos ( x ) + 1}{2} - cos (x) : \frac{1 - 3 cos ( x )}{2}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Add similar elements:

- cos (x) - 2 cos (x) = - 3 cos (x)

= \frac{1 - 3 cos ( x )}{2}

= \frac{- 3 cos ( x ) + 1}{2} + 1

= \frac{1 - 3 cos ( x )}{2} + 1

1 + \frac{1 - 3 cos ( x )}{2} = 0

Move

1

to the right side

1 + \frac{1 - 3 cos ( x )}{2} = 0

Subtract

1

from both sides

1 + \frac{1 - 3 cos ( x )}{2} - 1 = 0 - 1

Simplify

\frac{1 - 3 cos ( x )}{2} = - 1

\frac{1 - 3 cos ( x )}{2} = - 1

Multiply both sides by

2

\frac{1 - 3 cos ( x )}{2} = - 1

Multiply both sides by

2

\frac{2 ( 1 - 3 cos ( x ) )}{2} = 2 (- 1)

Simplify

1 - 3 cos (x) = - 2

1 - 3 cos (x) = - 2

Move

1

to the right side

1 - 3 cos (x) = - 2

Subtract

1

from both sides

1 - 3 cos (x) - 1 = - 2 - 1

Simplify

- 3 cos (x) = - 3

- 3 cos (x) = - 3

Divide both sides by

- 3

- 3 cos (x) = - 3

Divide both sides by

- 3

\frac{- 3 cos ( x )}{- 3} = \frac{- 3}{- 3}

Simplify

cos (x) = 1

cos (x) = 1

General solutions for

cos (x) = 1

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 = 0 + 2 πn

x = 0 + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

Popular Examples

sin(x)=cos(2x),0<= x<= 2pi

sin (x) = cos (2 x), 0 \leq x \leq 2 π

sin(2x)=sin(x)cos(x)

sin (2 x) = sin (x) cos (x)

sin(θ)=-(sqrt(5))/5 ,(3pi)/2 <θ<2pi

sin (θ) = - \frac{5}{5}, \frac{3 π}{2} < θ < 2 π

(2sin(2x)-1)(3tan(2x)-1)=0

(2 sin (2 x) - 1) (3 tan (2 x) - 1) = 0

-0.2588+0.9659tan(b)=1.4201

- 0.2588 + 0.9659 tan (b) = 1.4201

Frequently Asked Questions (FAQ)

What is the general solution for sin^2(x/2)=cos(x)-1 ?
The general solution for sin^2(x/2)=cos(x)-1 is x=2pin