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

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

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

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

x = 2 πn, x = π + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

1 + sin (x)

from both sides

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

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

Simplify

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

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

= cos (x) - sin (x) - cos (x)

Add similar elements:

cos (x) - cos (x) = 0

= - sin (x)

= - sin (x)

- sin (x) = 0

Divide both sides by

- 1

- sin (x) = 0

Divide both sides by

- 1

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

Simplify

sin (x) = 0

sin (x) = 0

General solutions for

sin (x) = 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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 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, x = π + 2 πn

5 sin^{2} (x) - 6 sin (x) cos (x) = 0

10 sin (20 t - \frac{π}{6}) = 8

x, - cos (x) = 0

sin^{2} (x) tan^{2} (x) = tan^{2} (x) + sin^{2} (x)

θ, cot (θ) = - \frac{π}{6}

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