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Mirrors > Home > MPE Home > Th. List > sincos3rdpi | Structured version Visualization version GIF version |
Description: The sine and cosine of π / 3. (Contributed by Mario Carneiro, 21-May-2016.) |
Ref | Expression |
---|---|
sincos3rdpi | ⊢ ((sin‘(π / 3)) = ((√‘3) / 2) ∧ (cos‘(π / 3)) = (1 / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | picn 25321 | . . . . . . 7 ⊢ π ∈ ℂ | |
2 | 2cn 11888 | . . . . . . . 8 ⊢ 2 ∈ ℂ | |
3 | 2ne0 11917 | . . . . . . . 8 ⊢ 2 ≠ 0 | |
4 | 2, 3 | reccli 11545 | . . . . . . 7 ⊢ (1 / 2) ∈ ℂ |
5 | 3cn 11894 | . . . . . . . 8 ⊢ 3 ∈ ℂ | |
6 | 3ne0 11919 | . . . . . . . 8 ⊢ 3 ≠ 0 | |
7 | 5, 6 | reccli 11545 | . . . . . . 7 ⊢ (1 / 3) ∈ ℂ |
8 | 1, 4, 7 | subdii 11264 | . . . . . 6 ⊢ (π · ((1 / 2) − (1 / 3))) = ((π · (1 / 2)) − (π · (1 / 3))) |
9 | halfthird 12419 | . . . . . . 7 ⊢ ((1 / 2) − (1 / 3)) = (1 / 6) | |
10 | 9 | oveq2i 7213 | . . . . . 6 ⊢ (π · ((1 / 2) − (1 / 3))) = (π · (1 / 6)) |
11 | 8, 10 | eqtr3i 2764 | . . . . 5 ⊢ ((π · (1 / 2)) − (π · (1 / 3))) = (π · (1 / 6)) |
12 | 1, 2, 3 | divreci 11560 | . . . . . 6 ⊢ (π / 2) = (π · (1 / 2)) |
13 | 1, 5, 6 | divreci 11560 | . . . . . 6 ⊢ (π / 3) = (π · (1 / 3)) |
14 | 12, 13 | oveq12i 7214 | . . . . 5 ⊢ ((π / 2) − (π / 3)) = ((π · (1 / 2)) − (π · (1 / 3))) |
15 | 6cn 11904 | . . . . . 6 ⊢ 6 ∈ ℂ | |
16 | 6nn 11902 | . . . . . . 7 ⊢ 6 ∈ ℕ | |
17 | 16 | nnne0i 11853 | . . . . . 6 ⊢ 6 ≠ 0 |
18 | 1, 15, 17 | divreci 11560 | . . . . 5 ⊢ (π / 6) = (π · (1 / 6)) |
19 | 11, 14, 18 | 3eqtr4i 2772 | . . . 4 ⊢ ((π / 2) − (π / 3)) = (π / 6) |
20 | 19 | fveq2i 6709 | . . 3 ⊢ (cos‘((π / 2) − (π / 3))) = (cos‘(π / 6)) |
21 | 1, 5, 6 | divcli 11557 | . . . 4 ⊢ (π / 3) ∈ ℂ |
22 | coshalfpim 25357 | . . . 4 ⊢ ((π / 3) ∈ ℂ → (cos‘((π / 2) − (π / 3))) = (sin‘(π / 3))) | |
23 | 21, 22 | ax-mp 5 | . . 3 ⊢ (cos‘((π / 2) − (π / 3))) = (sin‘(π / 3)) |
24 | sincos6thpi 25377 | . . . 4 ⊢ ((sin‘(π / 6)) = (1 / 2) ∧ (cos‘(π / 6)) = ((√‘3) / 2)) | |
25 | 24 | simpri 489 | . . 3 ⊢ (cos‘(π / 6)) = ((√‘3) / 2) |
26 | 20, 23, 25 | 3eqtr3i 2770 | . 2 ⊢ (sin‘(π / 3)) = ((√‘3) / 2) |
27 | 19 | fveq2i 6709 | . . 3 ⊢ (sin‘((π / 2) − (π / 3))) = (sin‘(π / 6)) |
28 | sinhalfpim 25355 | . . . 4 ⊢ ((π / 3) ∈ ℂ → (sin‘((π / 2) − (π / 3))) = (cos‘(π / 3))) | |
29 | 21, 28 | ax-mp 5 | . . 3 ⊢ (sin‘((π / 2) − (π / 3))) = (cos‘(π / 3)) |
30 | 24 | simpli 487 | . . 3 ⊢ (sin‘(π / 6)) = (1 / 2) |
31 | 27, 29, 30 | 3eqtr3i 2770 | . 2 ⊢ (cos‘(π / 3)) = (1 / 2) |
32 | 26, 31 | pm3.2i 474 | 1 ⊢ ((sin‘(π / 3)) = ((√‘3) / 2) ∧ (cos‘(π / 3)) = (1 / 2)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ∈ wcel 2110 ‘cfv 6369 (class class class)co 7202 ℂcc 10710 1c1 10713 · cmul 10717 − cmin 11045 / cdiv 11472 2c2 11868 3c3 11869 6c6 11872 √csqrt 14779 sincsin 15606 cosccos 15607 πcpi 15609 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 ax-addf 10791 ax-mulf 10792 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-iin 4897 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-2o 8192 df-er 8380 df-map 8499 df-pm 8500 df-ixp 8568 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-fi 9016 df-sup 9047 df-inf 9048 df-oi 9115 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-q 12528 df-rp 12570 df-xneg 12687 df-xadd 12688 df-xmul 12689 df-ioo 12922 df-ioc 12923 df-ico 12924 df-icc 12925 df-fz 13079 df-fzo 13222 df-fl 13350 df-seq 13558 df-exp 13619 df-fac 13823 df-bc 13852 df-hash 13880 df-shft 14613 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-limsup 15015 df-clim 15032 df-rlim 15033 df-sum 15233 df-ef 15610 df-sin 15612 df-cos 15613 df-pi 15615 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-starv 16782 df-sca 16783 df-vsca 16784 df-ip 16785 df-tset 16786 df-ple 16787 df-ds 16789 df-unif 16790 df-hom 16791 df-cco 16792 df-rest 16899 df-topn 16900 df-0g 16918 df-gsum 16919 df-topgen 16920 df-pt 16921 df-prds 16924 df-xrs 16979 df-qtop 16984 df-imas 16985 df-xps 16987 df-mre 17061 df-mrc 17062 df-acs 17064 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-submnd 18191 df-mulg 18461 df-cntz 18683 df-cmn 19144 df-psmet 20327 df-xmet 20328 df-met 20329 df-bl 20330 df-mopn 20331 df-fbas 20332 df-fg 20333 df-cnfld 20336 df-top 21763 df-topon 21780 df-topsp 21802 df-bases 21815 df-cld 21888 df-ntr 21889 df-cls 21890 df-nei 21967 df-lp 22005 df-perf 22006 df-cn 22096 df-cnp 22097 df-haus 22184 df-tx 22431 df-hmeo 22624 df-fil 22715 df-fm 22807 df-flim 22808 df-flf 22809 df-xms 23190 df-ms 23191 df-tms 23192 df-cncf 23747 df-limc 24735 df-dv 24736 |
This theorem is referenced by: pige3ALT 25381 acos1half 39844 |
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