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Mirrors > Home > MPE Home > Th. List > sincosq1eq | Structured version Visualization version GIF version |
Description: Complementarity of the sine and cosine functions in the first quadrant. (Contributed by Paul Chapman, 25-Jan-2008.) |
Ref | Expression |
---|---|
sincosq1eq | β’ ((π΄ β β β§ π΅ β β β§ (π΄ + π΅) = 1) β (sinβ(π΄ Β· (Ο / 2))) = (cosβ(π΅ Β· (Ο / 2)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | picn 25839 | . . . . . 6 β’ Ο β β | |
2 | 2cn 12236 | . . . . . 6 β’ 2 β β | |
3 | 2ne0 12265 | . . . . . 6 β’ 2 β 0 | |
4 | 1, 2, 3 | divcli 11905 | . . . . 5 β’ (Ο / 2) β β |
5 | mulcl 11143 | . . . . 5 β’ ((π΄ β β β§ (Ο / 2) β β) β (π΄ Β· (Ο / 2)) β β) | |
6 | 4, 5 | mpan2 690 | . . . 4 β’ (π΄ β β β (π΄ Β· (Ο / 2)) β β) |
7 | coshalfpim 25875 | . . . 4 β’ ((π΄ Β· (Ο / 2)) β β β (cosβ((Ο / 2) β (π΄ Β· (Ο / 2)))) = (sinβ(π΄ Β· (Ο / 2)))) | |
8 | 6, 7 | syl 17 | . . 3 β’ (π΄ β β β (cosβ((Ο / 2) β (π΄ Β· (Ο / 2)))) = (sinβ(π΄ Β· (Ο / 2)))) |
9 | 8 | 3ad2ant1 1134 | . 2 β’ ((π΄ β β β§ π΅ β β β§ (π΄ + π΅) = 1) β (cosβ((Ο / 2) β (π΄ Β· (Ο / 2)))) = (sinβ(π΄ Β· (Ο / 2)))) |
10 | adddir 11154 | . . . . . . 7 β’ ((π΄ β β β§ π΅ β β β§ (Ο / 2) β β) β ((π΄ + π΅) Β· (Ο / 2)) = ((π΄ Β· (Ο / 2)) + (π΅ Β· (Ο / 2)))) | |
11 | 4, 10 | mp3an3 1451 | . . . . . 6 β’ ((π΄ β β β§ π΅ β β) β ((π΄ + π΅) Β· (Ο / 2)) = ((π΄ Β· (Ο / 2)) + (π΅ Β· (Ο / 2)))) |
12 | 11 | 3adant3 1133 | . . . . 5 β’ ((π΄ β β β§ π΅ β β β§ (π΄ + π΅) = 1) β ((π΄ + π΅) Β· (Ο / 2)) = ((π΄ Β· (Ο / 2)) + (π΅ Β· (Ο / 2)))) |
13 | oveq1 7368 | . . . . . . 7 β’ ((π΄ + π΅) = 1 β ((π΄ + π΅) Β· (Ο / 2)) = (1 Β· (Ο / 2))) | |
14 | 4 | mulid2i 11168 | . . . . . . 7 β’ (1 Β· (Ο / 2)) = (Ο / 2) |
15 | 13, 14 | eqtrdi 2789 | . . . . . 6 β’ ((π΄ + π΅) = 1 β ((π΄ + π΅) Β· (Ο / 2)) = (Ο / 2)) |
16 | 15 | 3ad2ant3 1136 | . . . . 5 β’ ((π΄ β β β§ π΅ β β β§ (π΄ + π΅) = 1) β ((π΄ + π΅) Β· (Ο / 2)) = (Ο / 2)) |
17 | 12, 16 | eqtr3d 2775 | . . . 4 β’ ((π΄ β β β§ π΅ β β β§ (π΄ + π΅) = 1) β ((π΄ Β· (Ο / 2)) + (π΅ Β· (Ο / 2))) = (Ο / 2)) |
18 | mulcl 11143 | . . . . . . 7 β’ ((π΅ β β β§ (Ο / 2) β β) β (π΅ Β· (Ο / 2)) β β) | |
19 | 4, 18 | mpan2 690 | . . . . . 6 β’ (π΅ β β β (π΅ Β· (Ο / 2)) β β) |
20 | subadd 11412 | . . . . . 6 β’ (((Ο / 2) β β β§ (π΄ Β· (Ο / 2)) β β β§ (π΅ Β· (Ο / 2)) β β) β (((Ο / 2) β (π΄ Β· (Ο / 2))) = (π΅ Β· (Ο / 2)) β ((π΄ Β· (Ο / 2)) + (π΅ Β· (Ο / 2))) = (Ο / 2))) | |
21 | 4, 6, 19, 20 | mp3an3an 1468 | . . . . 5 β’ ((π΄ β β β§ π΅ β β) β (((Ο / 2) β (π΄ Β· (Ο / 2))) = (π΅ Β· (Ο / 2)) β ((π΄ Β· (Ο / 2)) + (π΅ Β· (Ο / 2))) = (Ο / 2))) |
22 | 21 | 3adant3 1133 | . . . 4 β’ ((π΄ β β β§ π΅ β β β§ (π΄ + π΅) = 1) β (((Ο / 2) β (π΄ Β· (Ο / 2))) = (π΅ Β· (Ο / 2)) β ((π΄ Β· (Ο / 2)) + (π΅ Β· (Ο / 2))) = (Ο / 2))) |
23 | 17, 22 | mpbird 257 | . . 3 β’ ((π΄ β β β§ π΅ β β β§ (π΄ + π΅) = 1) β ((Ο / 2) β (π΄ Β· (Ο / 2))) = (π΅ Β· (Ο / 2))) |
24 | 23 | fveq2d 6850 | . 2 β’ ((π΄ β β β§ π΅ β β β§ (π΄ + π΅) = 1) β (cosβ((Ο / 2) β (π΄ Β· (Ο / 2)))) = (cosβ(π΅ Β· (Ο / 2)))) |
25 | 9, 24 | eqtr3d 2775 | 1 β’ ((π΄ β β β§ π΅ β β β§ (π΄ + π΅) = 1) β (sinβ(π΄ Β· (Ο / 2))) = (cosβ(π΅ Β· (Ο / 2)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ w3a 1088 = wceq 1542 β wcel 2107 βcfv 6500 (class class class)co 7361 βcc 11057 1c1 11060 + caddc 11062 Β· cmul 11064 β cmin 11393 / cdiv 11820 2c2 12216 sincsin 15954 cosccos 15955 Οcpi 15957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-inf2 9585 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 ax-addf 11138 ax-mulf 11139 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-er 8654 df-map 8773 df-pm 8774 df-ixp 8842 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-fi 9355 df-sup 9386 df-inf 9387 df-oi 9454 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-q 12882 df-rp 12924 df-xneg 13041 df-xadd 13042 df-xmul 13043 df-ioo 13277 df-ioc 13278 df-ico 13279 df-icc 13280 df-fz 13434 df-fzo 13577 df-fl 13706 df-seq 13916 df-exp 13977 df-fac 14183 df-bc 14212 df-hash 14240 df-shft 14961 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-limsup 15362 df-clim 15379 df-rlim 15380 df-sum 15580 df-ef 15958 df-sin 15960 df-cos 15961 df-pi 15963 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-starv 17156 df-sca 17157 df-vsca 17158 df-ip 17159 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-hom 17165 df-cco 17166 df-rest 17312 df-topn 17313 df-0g 17331 df-gsum 17332 df-topgen 17333 df-pt 17334 df-prds 17337 df-xrs 17392 df-qtop 17397 df-imas 17398 df-xps 17400 df-mre 17474 df-mrc 17475 df-acs 17477 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-submnd 18610 df-mulg 18881 df-cntz 19105 df-cmn 19572 df-psmet 20811 df-xmet 20812 df-met 20813 df-bl 20814 df-mopn 20815 df-fbas 20816 df-fg 20817 df-cnfld 20820 df-top 22266 df-topon 22283 df-topsp 22305 df-bases 22319 df-cld 22393 df-ntr 22394 df-cls 22395 df-nei 22472 df-lp 22510 df-perf 22511 df-cn 22601 df-cnp 22602 df-haus 22689 df-tx 22936 df-hmeo 23129 df-fil 23220 df-fm 23312 df-flim 23313 df-flf 23314 df-xms 23696 df-ms 23697 df-tms 23698 df-cncf 24264 df-limc 25253 df-dv 25254 |
This theorem is referenced by: sincos4thpi 25893 sincos6thpi 25895 |
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