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| Mirrors > Home > MPE Home > Th. List > Mathboxes > tan3rdpi | Structured version Visualization version GIF version | ||
| Description: The tangent of π / 3 is √3. (Contributed by SN, 2-Sep-2025.) |
| Ref | Expression |
|---|---|
| tan3rdpi | ⊢ (tan‘(π / 3)) = (√‘3) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | picn 26406 | . . . 4 ⊢ π ∈ ℂ | |
| 2 | 3cn 12314 | . . . 4 ⊢ 3 ∈ ℂ | |
| 3 | 3ne0 12339 | . . . 4 ⊢ 3 ≠ 0 | |
| 4 | 1, 2, 3 | divcli 11976 | . . 3 ⊢ (π / 3) ∈ ℂ |
| 5 | sincos3rdpi 26464 | . . . . 5 ⊢ ((sin‘(π / 3)) = ((√‘3) / 2) ∧ (cos‘(π / 3)) = (1 / 2)) | |
| 6 | 5 | simpri 485 | . . . 4 ⊢ (cos‘(π / 3)) = (1 / 2) |
| 7 | 0re 11230 | . . . . 5 ⊢ 0 ∈ ℝ | |
| 8 | halfgt0 12449 | . . . . 5 ⊢ 0 < (1 / 2) | |
| 9 | 7, 8 | gtneii 11340 | . . . 4 ⊢ (1 / 2) ≠ 0 |
| 10 | 6, 9 | eqnetri 3001 | . . 3 ⊢ (cos‘(π / 3)) ≠ 0 |
| 11 | tanval 16133 | . . 3 ⊢ (((π / 3) ∈ ℂ ∧ (cos‘(π / 3)) ≠ 0) → (tan‘(π / 3)) = ((sin‘(π / 3)) / (cos‘(π / 3)))) | |
| 12 | 4, 10, 11 | mp2an 692 | . 2 ⊢ (tan‘(π / 3)) = ((sin‘(π / 3)) / (cos‘(π / 3))) |
| 13 | 5 | simpli 483 | . . 3 ⊢ (sin‘(π / 3)) = ((√‘3) / 2) |
| 14 | 13, 6 | oveq12i 7412 | . 2 ⊢ ((sin‘(π / 3)) / (cos‘(π / 3))) = (((√‘3) / 2) / (1 / 2)) |
| 15 | 2 | a1i 11 | . . . . . 6 ⊢ (⊤ → 3 ∈ ℂ) |
| 16 | 15 | sqrtcld 15445 | . . . . 5 ⊢ (⊤ → (√‘3) ∈ ℂ) |
| 17 | 1cnd 11223 | . . . . 5 ⊢ (⊤ → 1 ∈ ℂ) | |
| 18 | 2cnd 12311 | . . . . 5 ⊢ (⊤ → 2 ∈ ℂ) | |
| 19 | ax-1ne0 11191 | . . . . . 6 ⊢ 1 ≠ 0 | |
| 20 | 19 | a1i 11 | . . . . 5 ⊢ (⊤ → 1 ≠ 0) |
| 21 | 2ne0 12337 | . . . . . 6 ⊢ 2 ≠ 0 | |
| 22 | 21 | a1i 11 | . . . . 5 ⊢ (⊤ → 2 ≠ 0) |
| 23 | 16, 17, 18, 20, 22 | divcan7d 12038 | . . . 4 ⊢ (⊤ → (((√‘3) / 2) / (1 / 2)) = ((√‘3) / 1)) |
| 24 | 16 | div1d 12002 | . . . 4 ⊢ (⊤ → ((√‘3) / 1) = (√‘3)) |
| 25 | 23, 24 | eqtrd 2769 | . . 3 ⊢ (⊤ → (((√‘3) / 2) / (1 / 2)) = (√‘3)) |
| 26 | 25 | mptru 1546 | . 2 ⊢ (((√‘3) / 2) / (1 / 2)) = (√‘3) |
| 27 | 12, 14, 26 | 3eqtri 2761 | 1 ⊢ (tan‘(π / 3)) = (√‘3) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1539 ⊤wtru 1540 ∈ wcel 2107 ≠ wne 2931 ‘cfv 6528 (class class class)co 7400 ℂcc 11120 0cc0 11122 1c1 11123 / cdiv 11887 2c2 12288 3c3 12289 √csqrt 15241 sincsin 16068 cosccos 16069 tanctan 16070 πcpi 16071 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5247 ax-sep 5264 ax-nul 5274 ax-pow 5333 ax-pr 5400 ax-un 7724 ax-inf2 9648 ax-cnex 11178 ax-resscn 11179 ax-1cn 11180 ax-icn 11181 ax-addcl 11182 ax-addrcl 11183 ax-mulcl 11184 ax-mulrcl 11185 ax-mulcom 11186 ax-addass 11187 ax-mulass 11188 ax-distr 11189 ax-i2m1 11190 ax-1ne0 11191 ax-1rid 11192 ax-rnegex 11193 ax-rrecex 11194 ax-cnre 11195 ax-pre-lttri 11196 ax-pre-lttrn 11197 ax-pre-ltadd 11198 ax-pre-mulgt0 11199 ax-pre-sup 11200 ax-addf 11201 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3357 df-reu 3358 df-rab 3414 df-v 3459 df-sbc 3764 df-csb 3873 df-dif 3927 df-un 3929 df-in 3931 df-ss 3941 df-pss 3944 df-nul 4307 df-if 4499 df-pw 4575 df-sn 4600 df-pr 4602 df-tp 4604 df-op 4606 df-uni 4882 df-int 4921 df-iun 4967 df-iin 4968 df-br 5118 df-opab 5180 df-mpt 5200 df-tr 5228 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6288 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6530 df-fn 6531 df-f 6532 df-f1 6533 df-fo 6534 df-f1o 6535 df-fv 6536 df-isom 6537 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7666 df-om 7857 df-1st 7983 df-2nd 7984 df-supp 8155 df-frecs 8275 df-wrecs 8306 df-recs 8380 df-rdg 8419 df-1o 8475 df-2o 8476 df-er 8714 df-map 8837 df-pm 8838 df-ixp 8907 df-en 8955 df-dom 8956 df-sdom 8957 df-fin 8958 df-fsupp 9369 df-fi 9418 df-sup 9449 df-inf 9450 df-oi 9517 df-card 9946 df-pnf 11264 df-mnf 11265 df-xr 11266 df-ltxr 11267 df-le 11268 df-sub 11461 df-neg 11462 df-div 11888 df-nn 12234 df-2 12296 df-3 12297 df-4 12298 df-5 12299 df-6 12300 df-7 12301 df-8 12302 df-9 12303 df-n0 12495 df-z 12582 df-dec 12702 df-uz 12846 df-q 12958 df-rp 13002 df-xneg 13121 df-xadd 13122 df-xmul 13123 df-ioo 13358 df-ioc 13359 df-ico 13360 df-icc 13361 df-fz 13515 df-fzo 13662 df-fl 13799 df-seq 14010 df-exp 14070 df-fac 14282 df-bc 14311 df-hash 14339 df-shft 15075 df-cj 15107 df-re 15108 df-im 15109 df-sqrt 15243 df-abs 15244 df-limsup 15476 df-clim 15493 df-rlim 15494 df-sum 15692 df-ef 16072 df-sin 16074 df-cos 16075 df-tan 16076 df-pi 16077 df-struct 17153 df-sets 17170 df-slot 17188 df-ndx 17200 df-base 17216 df-ress 17239 df-plusg 17271 df-mulr 17272 df-starv 17273 df-sca 17274 df-vsca 17275 df-ip 17276 df-tset 17277 df-ple 17278 df-ds 17280 df-unif 17281 df-hom 17282 df-cco 17283 df-rest 17423 df-topn 17424 df-0g 17442 df-gsum 17443 df-topgen 17444 df-pt 17445 df-prds 17448 df-xrs 17503 df-qtop 17508 df-imas 17509 df-xps 17511 df-mre 17585 df-mrc 17586 df-acs 17588 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-submnd 18749 df-mulg 19038 df-cntz 19287 df-cmn 19750 df-psmet 21294 df-xmet 21295 df-met 21296 df-bl 21297 df-mopn 21298 df-fbas 21299 df-fg 21300 df-cnfld 21303 df-top 22819 df-topon 22836 df-topsp 22858 df-bases 22871 df-cld 22944 df-ntr 22945 df-cls 22946 df-nei 23023 df-lp 23061 df-perf 23062 df-cn 23152 df-cnp 23153 df-haus 23240 df-tx 23487 df-hmeo 23680 df-fil 23771 df-fm 23863 df-flim 23864 df-flf 23865 df-xms 24246 df-ms 24247 df-tms 24248 df-cncf 24809 df-limc 25806 df-dv 25807 |
| This theorem is referenced by: (None) |
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