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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 14922 | . . . . . . 7 ⊢ π ∈ ℂ | |
2 | 2cn 9053 | . . . . . . . 8 ⊢ 2 ∈ ℂ | |
3 | 2ap0 9075 | . . . . . . . 8 ⊢ 2 # 0 | |
4 | 2, 3 | recclapi 8761 | . . . . . . 7 ⊢ (1 / 2) ∈ ℂ |
5 | 3cn 9057 | . . . . . . . 8 ⊢ 3 ∈ ℂ | |
6 | 3ap0 9078 | . . . . . . . 8 ⊢ 3 # 0 | |
7 | 5, 6 | recclapi 8761 | . . . . . . 7 ⊢ (1 / 3) ∈ ℂ |
8 | 1, 4, 7 | subdii 8426 | . . . . . 6 ⊢ (π · ((1 / 2) − (1 / 3))) = ((π · (1 / 2)) − (π · (1 / 3))) |
9 | halfthird 9590 | . . . . . . 7 ⊢ ((1 / 2) − (1 / 3)) = (1 / 6) | |
10 | 9 | oveq2i 5929 | . . . . . 6 ⊢ (π · ((1 / 2) − (1 / 3))) = (π · (1 / 6)) |
11 | 8, 10 | eqtr3i 2216 | . . . . 5 ⊢ ((π · (1 / 2)) − (π · (1 / 3))) = (π · (1 / 6)) |
12 | 1, 2, 3 | divrecapi 8776 | . . . . . 6 ⊢ (π / 2) = (π · (1 / 2)) |
13 | 1, 5, 6 | divrecapi 8776 | . . . . . 6 ⊢ (π / 3) = (π · (1 / 3)) |
14 | 12, 13 | oveq12i 5930 | . . . . 5 ⊢ ((π / 2) − (π / 3)) = ((π · (1 / 2)) − (π · (1 / 3))) |
15 | 6cn 9064 | . . . . . 6 ⊢ 6 ∈ ℂ | |
16 | 6nn 9147 | . . . . . . 7 ⊢ 6 ∈ ℕ | |
17 | 16 | nnap0i 9013 | . . . . . 6 ⊢ 6 # 0 |
18 | 1, 15, 17 | divrecapi 8776 | . . . . 5 ⊢ (π / 6) = (π · (1 / 6)) |
19 | 11, 14, 18 | 3eqtr4i 2224 | . . . 4 ⊢ ((π / 2) − (π / 3)) = (π / 6) |
20 | 19 | fveq2i 5557 | . . 3 ⊢ (cos‘((π / 2) − (π / 3))) = (cos‘(π / 6)) |
21 | 1, 5, 6 | divclapi 8773 | . . . 4 ⊢ (π / 3) ∈ ℂ |
22 | coshalfpim 14958 | . . . 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 14977 | . . . 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 2222 | . 2 ⊢ (sin‘(π / 3)) = ((√‘3) / 2) |
27 | 19 | fveq2i 5557 | . . 3 ⊢ (sin‘((π / 2) − (π / 3))) = (sin‘(π / 6)) |
28 | sinhalfpim 14956 | . . . 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 2222 | . 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 2164 ‘cfv 5254 (class class class)co 5918 ℂcc 7870 1c1 7873 · cmul 7877 − cmin 8190 / cdiv 8691 2c2 9033 3c3 9034 6c6 9037 √csqrt 11140 sincsin 11787 cosccos 11788 πcpi 11790 |
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 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 ax-arch 7991 ax-caucvg 7992 ax-pre-suploc 7993 ax-addf 7994 ax-mulf 7995 |
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 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-disj 4007 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-isom 5263 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-of 6130 df-1st 6193 df-2nd 6194 df-recs 6358 df-irdg 6423 df-frec 6444 df-1o 6469 df-oadd 6473 df-er 6587 df-map 6704 df-pm 6705 df-en 6795 df-dom 6796 df-fin 6797 df-sup 7043 df-inf 7044 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-2 9041 df-3 9042 df-4 9043 df-5 9044 df-6 9045 df-7 9046 df-8 9047 df-9 9048 df-n0 9241 df-z 9318 df-uz 9593 df-q 9685 df-rp 9720 df-xneg 9838 df-xadd 9839 df-ioo 9958 df-ioc 9959 df-ico 9960 df-icc 9961 df-fz 10075 df-fzo 10209 df-seqfrec 10519 df-exp 10610 df-fac 10797 df-bc 10819 df-ihash 10847 df-shft 10959 df-cj 10986 df-re 10987 df-im 10988 df-rsqrt 11142 df-abs 11143 df-clim 11422 df-sumdc 11497 df-ef 11791 df-sin 11793 df-cos 11794 df-pi 11796 df-rest 12852 df-topgen 12871 df-psmet 14039 df-xmet 14040 df-met 14041 df-bl 14042 df-mopn 14043 df-top 14166 df-topon 14179 df-bases 14211 df-ntr 14264 df-cn 14356 df-cnp 14357 df-tx 14421 df-cncf 14726 df-limced 14810 df-dvap 14811 |
This theorem is referenced by: (None) |
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