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Mirrors > Home > MPE Home > Th. List > cosasin | Structured version Visualization version GIF version |
Description: The cosine of the arcsine of 𝐴 is √(1 − 𝐴↑2). (Contributed by Mario Carneiro, 2-Apr-2015.) |
Ref | Expression |
---|---|
cosasin | ⊢ (𝐴 ∈ ℂ → (cos‘(arcsin‘𝐴)) = (√‘(1 − (𝐴↑2)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | asincl 25443 | . . 3 ⊢ (𝐴 ∈ ℂ → (arcsin‘𝐴) ∈ ℂ) | |
2 | cosval 15468 | . . 3 ⊢ ((arcsin‘𝐴) ∈ ℂ → (cos‘(arcsin‘𝐴)) = (((exp‘(i · (arcsin‘𝐴))) + (exp‘(-i · (arcsin‘𝐴)))) / 2)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ ℂ → (cos‘(arcsin‘𝐴)) = (((exp‘(i · (arcsin‘𝐴))) + (exp‘(-i · (arcsin‘𝐴)))) / 2)) |
4 | ax-1cn 10587 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
5 | sqcl 13476 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ) | |
6 | subcl 10877 | . . . . . . 7 ⊢ ((1 ∈ ℂ ∧ (𝐴↑2) ∈ ℂ) → (1 − (𝐴↑2)) ∈ ℂ) | |
7 | 4, 5, 6 | sylancr 589 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (1 − (𝐴↑2)) ∈ ℂ) |
8 | 7 | sqrtcld 14789 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (√‘(1 − (𝐴↑2))) ∈ ℂ) |
9 | ax-icn 10588 | . . . . . 6 ⊢ i ∈ ℂ | |
10 | mulcl 10613 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
11 | 9, 10 | mpan 688 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ) |
12 | 8, 11, 8 | ppncand 11029 | . . . 4 ⊢ (𝐴 ∈ ℂ → (((√‘(1 − (𝐴↑2))) + (i · 𝐴)) + ((√‘(1 − (𝐴↑2))) − (i · 𝐴))) = ((√‘(1 − (𝐴↑2))) + (√‘(1 − (𝐴↑2))))) |
13 | efiasin 25458 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘(i · (arcsin‘𝐴))) = ((i · 𝐴) + (√‘(1 − (𝐴↑2))))) | |
14 | 11, 8, 13 | comraddd 10846 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(i · (arcsin‘𝐴))) = ((√‘(1 − (𝐴↑2))) + (i · 𝐴))) |
15 | mulneg12 11070 | . . . . . . . . . 10 ⊢ ((i ∈ ℂ ∧ (arcsin‘𝐴) ∈ ℂ) → (-i · (arcsin‘𝐴)) = (i · -(arcsin‘𝐴))) | |
16 | 9, 1, 15 | sylancr 589 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (-i · (arcsin‘𝐴)) = (i · -(arcsin‘𝐴))) |
17 | asinneg 25456 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (arcsin‘-𝐴) = -(arcsin‘𝐴)) | |
18 | 17 | oveq2d 7164 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (i · (arcsin‘-𝐴)) = (i · -(arcsin‘𝐴))) |
19 | 16, 18 | eqtr4d 2857 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (-i · (arcsin‘𝐴)) = (i · (arcsin‘-𝐴))) |
20 | 19 | fveq2d 6667 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · (arcsin‘𝐴))) = (exp‘(i · (arcsin‘-𝐴)))) |
21 | negcl 10878 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
22 | efiasin 25458 | . . . . . . . 8 ⊢ (-𝐴 ∈ ℂ → (exp‘(i · (arcsin‘-𝐴))) = ((i · -𝐴) + (√‘(1 − (-𝐴↑2))))) | |
23 | 21, 22 | syl 17 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (exp‘(i · (arcsin‘-𝐴))) = ((i · -𝐴) + (√‘(1 − (-𝐴↑2))))) |
24 | mulneg2 11069 | . . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · -𝐴) = -(i · 𝐴)) | |
25 | 9, 24 | mpan 688 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (i · -𝐴) = -(i · 𝐴)) |
26 | sqneg 13474 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (-𝐴↑2) = (𝐴↑2)) | |
27 | 26 | oveq2d 7164 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (1 − (-𝐴↑2)) = (1 − (𝐴↑2))) |
28 | 27 | fveq2d 6667 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (√‘(1 − (-𝐴↑2))) = (√‘(1 − (𝐴↑2)))) |
29 | 25, 28 | oveq12d 7166 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((i · -𝐴) + (√‘(1 − (-𝐴↑2)))) = (-(i · 𝐴) + (√‘(1 − (𝐴↑2))))) |
30 | 20, 23, 29 | 3eqtrd 2858 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · (arcsin‘𝐴))) = (-(i · 𝐴) + (√‘(1 − (𝐴↑2))))) |
31 | 11 | negcld 10976 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → -(i · 𝐴) ∈ ℂ) |
32 | 31, 8 | addcomd 10834 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (-(i · 𝐴) + (√‘(1 − (𝐴↑2)))) = ((√‘(1 − (𝐴↑2))) + -(i · 𝐴))) |
33 | 8, 11 | negsubd 10995 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((√‘(1 − (𝐴↑2))) + -(i · 𝐴)) = ((√‘(1 − (𝐴↑2))) − (i · 𝐴))) |
34 | 30, 32, 33 | 3eqtrd 2858 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · (arcsin‘𝐴))) = ((√‘(1 − (𝐴↑2))) − (i · 𝐴))) |
35 | 14, 34 | oveq12d 7166 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · (arcsin‘𝐴))) + (exp‘(-i · (arcsin‘𝐴)))) = (((√‘(1 − (𝐴↑2))) + (i · 𝐴)) + ((√‘(1 − (𝐴↑2))) − (i · 𝐴)))) |
36 | 8 | 2timesd 11872 | . . . 4 ⊢ (𝐴 ∈ ℂ → (2 · (√‘(1 − (𝐴↑2)))) = ((√‘(1 − (𝐴↑2))) + (√‘(1 − (𝐴↑2))))) |
37 | 12, 35, 36 | 3eqtr4d 2864 | . . 3 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · (arcsin‘𝐴))) + (exp‘(-i · (arcsin‘𝐴)))) = (2 · (√‘(1 − (𝐴↑2))))) |
38 | 37 | oveq1d 7163 | . 2 ⊢ (𝐴 ∈ ℂ → (((exp‘(i · (arcsin‘𝐴))) + (exp‘(-i · (arcsin‘𝐴)))) / 2) = ((2 · (√‘(1 − (𝐴↑2)))) / 2)) |
39 | 2cnd 11707 | . . 3 ⊢ (𝐴 ∈ ℂ → 2 ∈ ℂ) | |
40 | 2ne0 11733 | . . . 4 ⊢ 2 ≠ 0 | |
41 | 40 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℂ → 2 ≠ 0) |
42 | 8, 39, 41 | divcan3d 11413 | . 2 ⊢ (𝐴 ∈ ℂ → ((2 · (√‘(1 − (𝐴↑2)))) / 2) = (√‘(1 − (𝐴↑2)))) |
43 | 3, 38, 42 | 3eqtrd 2858 | 1 ⊢ (𝐴 ∈ ℂ → (cos‘(arcsin‘𝐴)) = (√‘(1 − (𝐴↑2)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 ‘cfv 6348 (class class class)co 7148 ℂcc 10527 0cc0 10529 1c1 10530 ici 10531 + caddc 10532 · cmul 10534 − cmin 10862 -cneg 10863 / cdiv 11289 2c2 11684 ↑cexp 13421 √csqrt 14584 expce 15407 cosccos 15410 arcsincasin 25432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-inf2 9096 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 ax-addf 10608 ax-mulf 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-om 7573 df-1st 7681 df-2nd 7682 df-supp 7823 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-er 8281 df-map 8400 df-pm 8401 df-ixp 8454 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-fsupp 8826 df-fi 8867 df-sup 8898 df-inf 8899 df-oi 8966 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-q 12341 df-rp 12382 df-xneg 12499 df-xadd 12500 df-xmul 12501 df-ioo 12734 df-ioc 12735 df-ico 12736 df-icc 12737 df-fz 12885 df-fzo 13026 df-fl 13154 df-mod 13230 df-seq 13362 df-exp 13422 df-fac 13626 df-bc 13655 df-hash 13683 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413 df-sin 15415 df-cos 15416 df-pi 15418 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20529 df-xmet 20530 df-met 20531 df-bl 20532 df-mopn 20533 df-fbas 20534 df-fg 20535 df-cnfld 20538 df-top 21494 df-topon 21511 df-topsp 21533 df-bases 21546 df-cld 21619 df-ntr 21620 df-cls 21621 df-nei 21698 df-lp 21736 df-perf 21737 df-cn 21827 df-cnp 21828 df-haus 21915 df-tx 22162 df-hmeo 22355 df-fil 22446 df-fm 22538 df-flim 22539 df-flf 22540 df-xms 22922 df-ms 22923 df-tms 22924 df-cncf 23478 df-limc 24456 df-dv 24457 df-log 25132 df-asin 25435 |
This theorem is referenced by: sinacos 25475 |
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