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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 26206 | . . . . . . 7 ⊢ π ∈ ℂ | |
2 | 2cn 12292 | . . . . . . . 8 ⊢ 2 ∈ ℂ | |
3 | 2ne0 12321 | . . . . . . . 8 ⊢ 2 ≠ 0 | |
4 | 2, 3 | reccli 11949 | . . . . . . 7 ⊢ (1 / 2) ∈ ℂ |
5 | 3cn 12298 | . . . . . . . 8 ⊢ 3 ∈ ℂ | |
6 | 3ne0 12323 | . . . . . . . 8 ⊢ 3 ≠ 0 | |
7 | 5, 6 | reccli 11949 | . . . . . . 7 ⊢ (1 / 3) ∈ ℂ |
8 | 1, 4, 7 | subdii 11668 | . . . . . 6 ⊢ (π · ((1 / 2) − (1 / 3))) = ((π · (1 / 2)) − (π · (1 / 3))) |
9 | halfthird 12825 | . . . . . . 7 ⊢ ((1 / 2) − (1 / 3)) = (1 / 6) | |
10 | 9 | oveq2i 7423 | . . . . . 6 ⊢ (π · ((1 / 2) − (1 / 3))) = (π · (1 / 6)) |
11 | 8, 10 | eqtr3i 2761 | . . . . 5 ⊢ ((π · (1 / 2)) − (π · (1 / 3))) = (π · (1 / 6)) |
12 | 1, 2, 3 | divreci 11964 | . . . . . 6 ⊢ (π / 2) = (π · (1 / 2)) |
13 | 1, 5, 6 | divreci 11964 | . . . . . 6 ⊢ (π / 3) = (π · (1 / 3)) |
14 | 12, 13 | oveq12i 7424 | . . . . 5 ⊢ ((π / 2) − (π / 3)) = ((π · (1 / 2)) − (π · (1 / 3))) |
15 | 6cn 12308 | . . . . . 6 ⊢ 6 ∈ ℂ | |
16 | 6nn 12306 | . . . . . . 7 ⊢ 6 ∈ ℕ | |
17 | 16 | nnne0i 12257 | . . . . . 6 ⊢ 6 ≠ 0 |
18 | 1, 15, 17 | divreci 11964 | . . . . 5 ⊢ (π / 6) = (π · (1 / 6)) |
19 | 11, 14, 18 | 3eqtr4i 2769 | . . . 4 ⊢ ((π / 2) − (π / 3)) = (π / 6) |
20 | 19 | fveq2i 6894 | . . 3 ⊢ (cos‘((π / 2) − (π / 3))) = (cos‘(π / 6)) |
21 | 1, 5, 6 | divcli 11961 | . . . 4 ⊢ (π / 3) ∈ ℂ |
22 | coshalfpim 26242 | . . . 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 26262 | . . . 4 ⊢ ((sin‘(π / 6)) = (1 / 2) ∧ (cos‘(π / 6)) = ((√‘3) / 2)) | |
25 | 24 | simpri 485 | . . 3 ⊢ (cos‘(π / 6)) = ((√‘3) / 2) |
26 | 20, 23, 25 | 3eqtr3i 2767 | . 2 ⊢ (sin‘(π / 3)) = ((√‘3) / 2) |
27 | 19 | fveq2i 6894 | . . 3 ⊢ (sin‘((π / 2) − (π / 3))) = (sin‘(π / 6)) |
28 | sinhalfpim 26240 | . . . 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 483 | . . 3 ⊢ (sin‘(π / 6)) = (1 / 2) |
31 | 27, 29, 30 | 3eqtr3i 2767 | . 2 ⊢ (cos‘(π / 3)) = (1 / 2) |
32 | 26, 31 | pm3.2i 470 | 1 ⊢ ((sin‘(π / 3)) = ((√‘3) / 2) ∧ (cos‘(π / 3)) = (1 / 2)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2105 ‘cfv 6543 (class class class)co 7412 ℂcc 11112 1c1 11115 · cmul 11119 − cmin 11449 / cdiv 11876 2c2 12272 3c3 12273 6c6 12276 √csqrt 15185 sincsin 16012 cosccos 16013 πcpi 16015 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9640 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 ax-addf 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8151 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-er 8707 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9366 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9509 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-ioo 13333 df-ioc 13334 df-ico 13335 df-icc 13336 df-fz 13490 df-fzo 13633 df-fl 13762 df-seq 13972 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15019 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-limsup 15420 df-clim 15437 df-rlim 15438 df-sum 15638 df-ef 16016 df-sin 16018 df-cos 16019 df-pi 16021 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-mulg 18988 df-cntz 19223 df-cmn 19692 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-fbas 21142 df-fg 21143 df-cnfld 21146 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-cld 22744 df-ntr 22745 df-cls 22746 df-nei 22823 df-lp 22861 df-perf 22862 df-cn 22952 df-cnp 22953 df-haus 23040 df-tx 23287 df-hmeo 23480 df-fil 23571 df-fm 23663 df-flim 23664 df-flf 23665 df-xms 24047 df-ms 24048 df-tms 24049 df-cncf 24619 df-limc 25616 df-dv 25617 |
This theorem is referenced by: pige3ALT 26266 acos1half 41718 |
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