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Definition df-sin 14582
Description: Define the sine function. (Contributed by NM, 14-Mar-2005.)
Assertion
Ref Expression
df-sin sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)))

Detailed syntax breakdown of Definition df-sin
StepHypRef Expression
1 csin 14576 . 2 class sin
2 vx . . 3 setvar 𝑥
3 cc 9787 . . 3 class
4 ci 9791 . . . . . . 7 class i
52cv 1473 . . . . . . 7 class 𝑥
6 cmul 9794 . . . . . . 7 class ·
74, 5, 6co 6524 . . . . . 6 class (i · 𝑥)
8 ce 14574 . . . . . 6 class exp
97, 8cfv 5787 . . . . 5 class (exp‘(i · 𝑥))
104cneg 10115 . . . . . . 7 class -i
1110, 5, 6co 6524 . . . . . 6 class (-i · 𝑥)
1211, 8cfv 5787 . . . . 5 class (exp‘(-i · 𝑥))
13 cmin 10114 . . . . 5 class
149, 12, 13co 6524 . . . 4 class ((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥)))
15 c2 10914 . . . . 5 class 2
1615, 4, 6co 6524 . . . 4 class (2 · i)
17 cdiv 10530 . . . 4 class /
1814, 16, 17co 6524 . . 3 class (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i))
192, 3, 18cmpt 4634 . 2 class (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)))
201, 19wceq 1474 1 wff sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)))
Colors of variables: wff setvar class
This definition is referenced by:  sinval  14634  sinf  14636  dvsincos  23462  sincn  23916
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