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

The general solution for 2cos(x/2)=cos(x)+1 is x=pi+4pin,x=3pi+4pin,x=4pin

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

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

x = π + 4 πn, x = 3 π + 4 πn, x = 4 πn

Solution steps

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

Subtract

cos (x) + 1

from both sides

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

Let:

u = \frac{x}{2}

2 cos (u) - cos (2 u) - 1 = 0

Rewrite using trig identities

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

Solve by substitution

cos (u) = 0, cos (u) = 1

cos (u) = 0 : u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn

cos (u) = 1 : u = 2 πn

Combine all the solutions

u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn, u = 2 πn

Substitute back

u = \frac{x}{2}

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

\frac{x}{2} = \frac{3 π}{2} + 2 πn : x = 3 π + 4 πn

\frac{x}{2} = 2 πn : x = 4 πn

x = π + 4 πn, x = 3 π + 4 πn, x = 4 πn

cos (x) = 100

1.58 \cdot sin (3 4^{\circ}) = 1 (sin (θ))

10 sin (x) + 4 = 10

4. 3^{2} = 3. 8^{2} + 6. 3^{2} + 2 (3.8) (6.3) cos (b)

cos^{2} (x) - 4 cos (x) + 4 = 0

What is the general solution for 2cos(x/2)=cos(x)+1 ?
The general solution for 2cos(x/2)=cos(x)+1 is x=pi+4pin,x=3pi+4pin,x=4pin