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

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

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

Solution

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

Subtract

sin (x) + C

from both sides

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

Let:

u = \frac{x}{2}

2 sin^{2} (u) - sin (2 u) = 0

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

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

Factor

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

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

Apply exponent rule:

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

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

= 2 sin (u) sin (u) - 2 sin (u) cos (u)

Factor out common term

2 sin (u)

= 2 sin (u) (sin (u) - cos (u))

2 sin (u) (sin (u) - cos (u)) = 0

Solving each part separately

sin (u) = 0 or sin (u) - cos (u) = 0

sin (u) = 0 : u = 2 πn, u = π + 2 πn

sin (u) = 0

General solutions for

sin (u) = 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}

u = 0 + 2 πn, u = π + 2 πn

u = 0 + 2 πn, u = π + 2 πn

Solve

u = 0 + 2 πn : u = 2 πn

u = 0 + 2 πn

0 + 2 πn = 2 πn

u = 2 πn

u = 2 πn, u = π + 2 πn

sin (u) - cos (u) = 0 : u = \frac{π}{4} + πn

sin (u) - cos (u) = 0

Rewrite using trig identities

sin (u) - cos (u) = 0

Divide both sides by

cos (u), cos (u) \neq = 0

\frac{sin ( u ) - cos ( u )}{cos ( u )} = \frac{0}{cos ( u )}

Simplify

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

Use the basic trigonometric identity:

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

tan (u) - 1 = 0

tan (u) - 1 = 0

Move

1

to the right side

tan (u) - 1 = 0

Add

1

to both sides

tan (u) - 1 + 1 = 0 + 1

Simplify

tan (u) = 1

tan (u) = 1

General solutions for

tan (u) = 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}

u = \frac{π}{4} + πn

u = \frac{π}{4} + πn

Combine all the solutions

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

Substitute back

u = \frac{x}{2}

\frac{x}{2} = 2 πn : x = 4 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

\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

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

Popular Examples

sin(a)= 3/5 ,cos(a)

sin (a) = \frac{3}{5}, cos (a)

cos(x)=cos(4x)

cos (x) = cos (4 x)

11sin(B)+13=4sin(B)+8

11 sin (B) + 13 = 4 sin (B) + 8

2=4cos(x)

2 = 4 cos (x)

sin(2x+pi)=0

sin (2 x + π) = 0

Frequently Asked Questions (FAQ)

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