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

The general solution for 2sin^2(1/2 x)+C=2cos(x/2)sin(x/2)+C is x=4pin,x=2pi+4pin,x= pi/2+2pin

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2sin^2(1/2 x)+C=2cos(x/2)sin(x/2)+C

Solution

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

Solution

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

Degrees

x = 0^{\circ} + 72 0^{\circ} n, x = 36 0^{\circ} + 72 0^{\circ} n, x = 9 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

2 cos (\frac{x}{2}) sin (\frac{x}{2}) + C

from both sides

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

Factor

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

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

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

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

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

Factor out common term

2 sin (\frac{x}{2})

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

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

Solving each part separately

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

sin (\frac{x}{2}) = 0 : x = 4 πn, x = 2 π + 4 πn

sin (\frac{x}{2}) = 0

General solutions for

sin (\frac{x}{2}) = 0

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{x}{2} = 0 + 2 πn, \frac{x}{2} = π + 2 πn

\frac{x}{2} = 0 + 2 πn, \frac{x}{2} = π + 2 πn

Solve

\frac{x}{2} = 0 + 2 πn : x = 4 πn

\frac{x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{x}{2} = 2 πn

Multiply both sides by

2

\frac{x}{2} = 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot 2 πn

Simplify

x = 4 πn

x = 4 πn

Solve

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

\frac{x}{2} = π + 2 πn

Multiply both sides by

2

\frac{x}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 π + 2 \cdot 2 πn

Simplify

x = 2 π + 4 πn

x = 2 π + 4 πn

x = 4 πn, x = 2 π + 4 πn

sin (\frac{x}{2}) - cos (\frac{x}{2}) = 0 : x = \frac{π}{2} + 2 πn

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

Rewrite using trig identities

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

Divide both sides by

cos (\frac{x}{2}), cos (\frac{x}{2}) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

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

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

Move

1

to the right side

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

Add

1

to both sides

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

Simplify

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

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

General solutions for

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

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

\frac{x}{2} = \frac{π}{4} + πn

\frac{x}{2} = \frac{π}{4} + πn

Solve

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

\frac{x}{2} = \frac{π}{4} + πn

Multiply both sides by

2

\frac{x}{2} = \frac{π}{4} + πn

Multiply both sides by

2

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

Simplify

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

2 \cdot \frac{π}{4} + 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}

= \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

Combine all the solutions

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

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Frequently Asked Questions (FAQ)

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