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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 14611 | . . . . . . 7 ⊢ π ∈ ℂ | |
2 | 2cn 9009 | . . . . . . . 8 ⊢ 2 ∈ ℂ | |
3 | 2ap0 9031 | . . . . . . . 8 ⊢ 2 # 0 | |
4 | 2, 3 | recclapi 8718 | . . . . . . 7 ⊢ (1 / 2) ∈ ℂ |
5 | 3cn 9013 | . . . . . . . 8 ⊢ 3 ∈ ℂ | |
6 | 3ap0 9034 | . . . . . . . 8 ⊢ 3 # 0 | |
7 | 5, 6 | recclapi 8718 | . . . . . . 7 ⊢ (1 / 3) ∈ ℂ |
8 | 1, 4, 7 | subdii 8383 | . . . . . 6 ⊢ (π · ((1 / 2) − (1 / 3))) = ((π · (1 / 2)) − (π · (1 / 3))) |
9 | halfthird 9545 | . . . . . . 7 ⊢ ((1 / 2) − (1 / 3)) = (1 / 6) | |
10 | 9 | oveq2i 5902 | . . . . . 6 ⊢ (π · ((1 / 2) − (1 / 3))) = (π · (1 / 6)) |
11 | 8, 10 | eqtr3i 2212 | . . . . 5 ⊢ ((π · (1 / 2)) − (π · (1 / 3))) = (π · (1 / 6)) |
12 | 1, 2, 3 | divrecapi 8733 | . . . . . 6 ⊢ (π / 2) = (π · (1 / 2)) |
13 | 1, 5, 6 | divrecapi 8733 | . . . . . 6 ⊢ (π / 3) = (π · (1 / 3)) |
14 | 12, 13 | oveq12i 5903 | . . . . 5 ⊢ ((π / 2) − (π / 3)) = ((π · (1 / 2)) − (π · (1 / 3))) |
15 | 6cn 9020 | . . . . . 6 ⊢ 6 ∈ ℂ | |
16 | 6nn 9103 | . . . . . . 7 ⊢ 6 ∈ ℕ | |
17 | 16 | nnap0i 8969 | . . . . . 6 ⊢ 6 # 0 |
18 | 1, 15, 17 | divrecapi 8733 | . . . . 5 ⊢ (π / 6) = (π · (1 / 6)) |
19 | 11, 14, 18 | 3eqtr4i 2220 | . . . 4 ⊢ ((π / 2) − (π / 3)) = (π / 6) |
20 | 19 | fveq2i 5533 | . . 3 ⊢ (cos‘((π / 2) − (π / 3))) = (cos‘(π / 6)) |
21 | 1, 5, 6 | divclapi 8730 | . . . 4 ⊢ (π / 3) ∈ ℂ |
22 | coshalfpim 14647 | . . . 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 14666 | . . . 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 2218 | . 2 ⊢ (sin‘(π / 3)) = ((√‘3) / 2) |
27 | 19 | fveq2i 5533 | . . 3 ⊢ (sin‘((π / 2) − (π / 3))) = (sin‘(π / 6)) |
28 | sinhalfpim 14645 | . . . 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 2218 | . 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 1364 ∈ wcel 2160 ‘cfv 5231 (class class class)co 5891 ℂcc 7828 1c1 7831 · cmul 7835 − cmin 8147 / cdiv 8648 2c2 8989 3c3 8990 6c6 8993 √csqrt 11024 sincsin 11671 cosccos 11672 πcpi 11674 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7921 ax-resscn 7922 ax-1cn 7923 ax-1re 7924 ax-icn 7925 ax-addcl 7926 ax-addrcl 7927 ax-mulcl 7928 ax-mulrcl 7929 ax-addcom 7930 ax-mulcom 7931 ax-addass 7932 ax-mulass 7933 ax-distr 7934 ax-i2m1 7935 ax-0lt1 7936 ax-1rid 7937 ax-0id 7938 ax-rnegex 7939 ax-precex 7940 ax-cnre 7941 ax-pre-ltirr 7942 ax-pre-ltwlin 7943 ax-pre-lttrn 7944 ax-pre-apti 7945 ax-pre-ltadd 7946 ax-pre-mulgt0 7947 ax-pre-mulext 7948 ax-arch 7949 ax-caucvg 7950 ax-pre-suploc 7951 ax-addf 7952 ax-mulf 7953 |
This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-disj 3996 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-po 4311 df-iso 4312 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-isom 5240 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-of 6101 df-1st 6159 df-2nd 6160 df-recs 6324 df-irdg 6389 df-frec 6410 df-1o 6435 df-oadd 6439 df-er 6553 df-map 6668 df-pm 6669 df-en 6759 df-dom 6760 df-fin 6761 df-sup 7002 df-inf 7003 df-pnf 8013 df-mnf 8014 df-xr 8015 df-ltxr 8016 df-le 8017 df-sub 8149 df-neg 8150 df-reap 8551 df-ap 8558 df-div 8649 df-inn 8939 df-2 8997 df-3 8998 df-4 8999 df-5 9000 df-6 9001 df-7 9002 df-8 9003 df-9 9004 df-n0 9196 df-z 9273 df-uz 9548 df-q 9639 df-rp 9673 df-xneg 9791 df-xadd 9792 df-ioo 9911 df-ioc 9912 df-ico 9913 df-icc 9914 df-fz 10028 df-fzo 10162 df-seqfrec 10465 df-exp 10539 df-fac 10725 df-bc 10747 df-ihash 10775 df-shft 10843 df-cj 10870 df-re 10871 df-im 10872 df-rsqrt 11026 df-abs 11027 df-clim 11306 df-sumdc 11381 df-ef 11675 df-sin 11677 df-cos 11678 df-pi 11680 df-rest 12718 df-topgen 12737 df-psmet 13823 df-xmet 13824 df-met 13825 df-bl 13826 df-mopn 13827 df-top 13901 df-topon 13914 df-bases 13946 df-ntr 13999 df-cn 14091 df-cnp 14092 df-tx 14156 df-cncf 14461 df-limced 14528 df-dvap 14529 |
This theorem is referenced by: (None) |
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