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Metamath Proof Explorer

Theorem sinper

Description: The sine function is periodic. (Contributed by Paul Chapman, 23-Jan-2008) (Revised by Mario Carneiro, 10-May-2014)

Ref Expression
Assertion sinper ⊢ A ∈ ℂ ∧ K ∈ ℤ → sin ⁡ A + K ⁢ 2 ⁢ π = sin ⁡ A

Proof

Step Hyp Ref Expression
1 sinval ⊢ A ∈ ℂ → sin ⁡ A = e i ⁢ A − e − i ⁢ A 2 ⁢ i
2 sinval ⊢ A + K ⁢ 2 ⁢ π ∈ ℂ → sin ⁡ A + K ⁢ 2 ⁢ π = e i ⁢ A + K ⁢ 2 ⁢ π − e − i ⁢ A + K ⁢ 2 ⁢ π 2 ⁢ i
3 1 2 sinperlem ⊢ A ∈ ℂ ∧ K ∈ ℤ → sin ⁡ A + K ⁢ 2 ⁢ π = sin ⁡ A