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Definition df-cos 14583
Description: Define the cosine function. (Contributed by NM, 14-Mar-2005.)
Assertion
Ref Expression
df-cos cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))

Detailed syntax breakdown of Definition df-cos
StepHypRef Expression
1 ccos 14577 . 2 class cos
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 caddc 9792 . . . . 5 class +
149, 12, 13co 6524 . . . 4 class ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥)))
15 c2 10914 . . . 4 class 2
16 cdiv 10530 . . . 4 class /
1714, 15, 16co 6524 . . 3 class (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2)
182, 3, 17cmpt 4634 . 2 class (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))
191, 18wceq 1474 1 wff cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))
Colors of variables: wff setvar class
This definition is referenced by:  cosval  14635  cosf  14637  dvsincos  23462  coscn  23917
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