What is the general solution for cos(θ/2)=-1 ?

The general solution for cos(θ/2)=-1 is θ=2pi+4pin

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cos(θ/2)=-1

cos (\frac{θ}{2}) = - 1

θ = 2 π + 4 πn

Degrees

θ = 36 0^{\circ} + 72 0^{\circ} n

Solution steps

cos (\frac{θ}{2}) = - 1

General solutions for

cos (\frac{θ}{2}) = - 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}

\frac{θ}{2} = π + 2 πn

\frac{θ}{2} = π + 2 πn

Solve

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

\frac{θ}{2} = π + 2 πn

Multiply both sides by

2

\frac{θ}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 θ}{2} = 2 π + 2 \cdot 2 πn

Simplify

θ = 2 π + 4 πn

θ = 2 π + 4 πn

θ = 2 π + 4 πn

2 sin (x) + \frac{3}{4} = \frac{3}{2} sin (x)

- 2 cos^{2} (θ) + sin (θ) + 5 = 4

12 sin^{2} (x) - 9 = 0

2 cos (x - 1) = 0

5 + 6 sin (x) = 0

What is the general solution for cos(θ/2)=-1 ?
The general solution for cos(θ/2)=-1 is θ=2pi+4pin