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Mirrors > Home > ILE Home > Th. List > sincos3rdpi | 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 14479 | . . . . . . 7 ⊢ π ∈ ℂ | |
2 | 2cn 9003 | . . . . . . . 8 ⊢ 2 ∈ ℂ | |
3 | 2ap0 9025 | . . . . . . . 8 ⊢ 2 # 0 | |
4 | 2, 3 | recclapi 8712 | . . . . . . 7 ⊢ (1 / 2) ∈ ℂ |
5 | 3cn 9007 | . . . . . . . 8 ⊢ 3 ∈ ℂ | |
6 | 3ap0 9028 | . . . . . . . 8 ⊢ 3 # 0 | |
7 | 5, 6 | recclapi 8712 | . . . . . . 7 ⊢ (1 / 3) ∈ ℂ |
8 | 1, 4, 7 | subdii 8377 | . . . . . 6 ⊢ (π · ((1 / 2) − (1 / 3))) = ((π · (1 / 2)) − (π · (1 / 3))) |
9 | halfthird 9539 | . . . . . . 7 ⊢ ((1 / 2) − (1 / 3)) = (1 / 6) | |
10 | 9 | oveq2i 5899 | . . . . . 6 ⊢ (π · ((1 / 2) − (1 / 3))) = (π · (1 / 6)) |
11 | 8, 10 | eqtr3i 2210 | . . . . 5 ⊢ ((π · (1 / 2)) − (π · (1 / 3))) = (π · (1 / 6)) |
12 | 1, 2, 3 | divrecapi 8727 | . . . . . 6 ⊢ (π / 2) = (π · (1 / 2)) |
13 | 1, 5, 6 | divrecapi 8727 | . . . . . 6 ⊢ (π / 3) = (π · (1 / 3)) |
14 | 12, 13 | oveq12i 5900 | . . . . 5 ⊢ ((π / 2) − (π / 3)) = ((π · (1 / 2)) − (π · (1 / 3))) |
15 | 6cn 9014 | . . . . . 6 ⊢ 6 ∈ ℂ | |
16 | 6nn 9097 | . . . . . . 7 ⊢ 6 ∈ ℕ | |
17 | 16 | nnap0i 8963 | . . . . . 6 ⊢ 6 # 0 |
18 | 1, 15, 17 | divrecapi 8727 | . . . . 5 ⊢ (π / 6) = (π · (1 / 6)) |
19 | 11, 14, 18 | 3eqtr4i 2218 | . . . 4 ⊢ ((π / 2) − (π / 3)) = (π / 6) |
20 | 19 | fveq2i 5530 | . . 3 ⊢ (cos‘((π / 2) − (π / 3))) = (cos‘(π / 6)) |
21 | 1, 5, 6 | divclapi 8724 | . . . 4 ⊢ (π / 3) ∈ ℂ |
22 | coshalfpim 14515 | . . . 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 14534 | . . . 4 ⊢ ((sin‘(π / 6)) = (1 / 2) ∧ (cos‘(π / 6)) = ((√‘3) / 2)) | |
25 | 24 | simpri 113 | . . 3 ⊢ (cos‘(π / 6)) = ((√‘3) / 2) |
26 | 20, 23, 25 | 3eqtr3i 2216 | . 2 ⊢ (sin‘(π / 3)) = ((√‘3) / 2) |
27 | 19 | fveq2i 5530 | . . 3 ⊢ (sin‘((π / 2) − (π / 3))) = (sin‘(π / 6)) |
28 | sinhalfpim 14513 | . . . 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 111 | . . 3 ⊢ (sin‘(π / 6)) = (1 / 2) |
31 | 27, 29, 30 | 3eqtr3i 2216 | . 2 ⊢ (cos‘(π / 3)) = (1 / 2) |
32 | 26, 31 | pm3.2i 272 | 1 ⊢ ((sin‘(π / 3)) = ((√‘3) / 2) ∧ (cos‘(π / 3)) = (1 / 2)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1363 ∈ wcel 2158 ‘cfv 5228 (class class class)co 5888 ℂcc 7822 1c1 7825 · cmul 7829 − cmin 8141 / cdiv 8642 2c2 8983 3c3 8984 6c6 8987 √csqrt 11018 sincsin 11665 cosccos 11666 πcpi 11668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-mulrcl 7923 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-precex 7934 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 ax-pre-mulgt0 7941 ax-pre-mulext 7942 ax-arch 7943 ax-caucvg 7944 ax-pre-suploc 7945 ax-addf 7946 ax-mulf 7947 |
This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-disj 3993 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-po 4308 df-iso 4309 df-iord 4378 df-on 4380 df-ilim 4381 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-isom 5237 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-of 6096 df-1st 6154 df-2nd 6155 df-recs 6319 df-irdg 6384 df-frec 6405 df-1o 6430 df-oadd 6434 df-er 6548 df-map 6663 df-pm 6664 df-en 6754 df-dom 6755 df-fin 6756 df-sup 6996 df-inf 6997 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-reap 8545 df-ap 8552 df-div 8643 df-inn 8933 df-2 8991 df-3 8992 df-4 8993 df-5 8994 df-6 8995 df-7 8996 df-8 8997 df-9 8998 df-n0 9190 df-z 9267 df-uz 9542 df-q 9633 df-rp 9667 df-xneg 9785 df-xadd 9786 df-ioo 9905 df-ioc 9906 df-ico 9907 df-icc 9908 df-fz 10022 df-fzo 10156 df-seqfrec 10459 df-exp 10533 df-fac 10719 df-bc 10741 df-ihash 10769 df-shft 10837 df-cj 10864 df-re 10865 df-im 10866 df-rsqrt 11020 df-abs 11021 df-clim 11300 df-sumdc 11375 df-ef 11669 df-sin 11671 df-cos 11672 df-pi 11674 df-rest 12707 df-topgen 12726 df-psmet 13704 df-xmet 13705 df-met 13706 df-bl 13707 df-mopn 13708 df-top 13769 df-topon 13782 df-bases 13814 df-ntr 13867 df-cn 13959 df-cnp 13960 df-tx 14024 df-cncf 14329 df-limced 14396 df-dvap 14397 |
This theorem is referenced by: (None) |
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