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Definition df-sin 14844
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 14838 . 2 class sin
2 vx . . 3 setvar 𝑥
3 cc 9972 . . 3 class
4 ci 9976 . . . . . . 7 class i
52cv 1522 . . . . . . 7 class 𝑥
6 cmul 9979 . . . . . . 7 class ·
74, 5, 6co 6690 . . . . . 6 class (i · 𝑥)
8 ce 14836 . . . . . 6 class exp
97, 8cfv 5926 . . . . 5 class (exp‘(i · 𝑥))
104cneg 10305 . . . . . . 7 class -i
1110, 5, 6co 6690 . . . . . 6 class (-i · 𝑥)
1211, 8cfv 5926 . . . . 5 class (exp‘(-i · 𝑥))
13 cmin 10304 . . . . 5 class
149, 12, 13co 6690 . . . 4 class ((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥)))
15 c2 11108 . . . . 5 class 2
1615, 4, 6co 6690 . . . 4 class (2 · i)
17 cdiv 10722 . . . 4 class /
1814, 16, 17co 6690 . . 3 class (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i))
192, 3, 18cmpt 4762 . 2 class (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)))
201, 19wceq 1523 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  14896  sinf  14898  dvsincos  23789  sincn  24243
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