cos θ = -1

How to find the general solution of an equation of the form cos θ = -1?

Prove that the general solution of cos θ = -1 is given by θ = (2n + 1)π, n ∈ Z.

Solution:

We have,

cos θ = -1       

⇒ cos θ = cos π

θ = 2mπ ± π, m ∈ Z, [Since, the general solution of cos θ = cos ∝ is given by θ = 2nπ ± ∝, n ∈ Z.]

⇒ θ = (2m ± 1)π, m ∈ Z, (i.e., n = 0, ± 1,± 2, …………)

⇒ θ = odd multiple of π = (2n + 1)π , where n ∈ Z,(i.e., n = 0, ± 1,± 2, …………)

Hence, the general solution of cos θ = -1 is θ = (2n + 1)π, n ∈ Z (i.e., n = 0, ± 1,± 2, …………)

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