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Mirrors > Home > ILE Home > Th. List > sincosq1eq | 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 | halfpire 14484 | . . . . . 6 β’ (Ο / 2) β β | |
2 | 1 | recni 7982 | . . . . 5 β’ (Ο / 2) β β |
3 | mulcl 7951 | . . . . 5 β’ ((π΄ β β β§ (Ο / 2) β β) β (π΄ Β· (Ο / 2)) β β) | |
4 | 2, 3 | mpan2 425 | . . . 4 β’ (π΄ β β β (π΄ Β· (Ο / 2)) β β) |
5 | coshalfpim 14515 | . . . 4 β’ ((π΄ Β· (Ο / 2)) β β β (cosβ((Ο / 2) β (π΄ Β· (Ο / 2)))) = (sinβ(π΄ Β· (Ο / 2)))) | |
6 | 4, 5 | syl 14 | . . 3 β’ (π΄ β β β (cosβ((Ο / 2) β (π΄ Β· (Ο / 2)))) = (sinβ(π΄ Β· (Ο / 2)))) |
7 | 6 | 3ad2ant1 1019 | . 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)))) |
8 | adddir 7961 | . . . . . . 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)))) | |
9 | 2, 8 | mp3an3 1336 | . . . . . 6 β’ ((π΄ β β β§ π΅ β β) β ((π΄ + π΅) Β· (Ο / 2)) = ((π΄ Β· (Ο / 2)) + (π΅ Β· (Ο / 2)))) |
10 | 9 | 3adant3 1018 | . . . . 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)))) |
11 | oveq1 5895 | . . . . . . 7 β’ ((π΄ + π΅) = 1 β ((π΄ + π΅) Β· (Ο / 2)) = (1 Β· (Ο / 2))) | |
12 | 2 | mullidi 7973 | . . . . . . 7 β’ (1 Β· (Ο / 2)) = (Ο / 2) |
13 | 11, 12 | eqtrdi 2236 | . . . . . 6 β’ ((π΄ + π΅) = 1 β ((π΄ + π΅) Β· (Ο / 2)) = (Ο / 2)) |
14 | 13 | 3ad2ant3 1021 | . . . . 5 β’ ((π΄ β β β§ π΅ β β β§ (π΄ + π΅) = 1) β ((π΄ + π΅) Β· (Ο / 2)) = (Ο / 2)) |
15 | 10, 14 | eqtr3d 2222 | . . . 4 β’ ((π΄ β β β§ π΅ β β β§ (π΄ + π΅) = 1) β ((π΄ Β· (Ο / 2)) + (π΅ Β· (Ο / 2))) = (Ο / 2)) |
16 | mulcl 7951 | . . . . . . 7 β’ ((π΅ β β β§ (Ο / 2) β β) β (π΅ Β· (Ο / 2)) β β) | |
17 | 2, 16 | mpan2 425 | . . . . . 6 β’ (π΅ β β β (π΅ Β· (Ο / 2)) β β) |
18 | subadd 8173 | . . . . . 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))) | |
19 | 2, 4, 17, 18 | mp3an3an 1353 | . . . . 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))) |
20 | 19 | 3adant3 1018 | . . . 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))) |
21 | 15, 20 | mpbird 167 | . . 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))) |
22 | 21 | fveq2d 5531 | . 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)))) |
23 | 7, 22 | eqtr3d 2222 | 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 set class |
Syntax hints: β wi 4 β wb 105 β§ w3a 979 = wceq 1363 β wcel 2158 βcfv 5228 (class class class)co 5888 βcc 7822 1c1 7825 + caddc 7827 Β· cmul 7829 β cmin 8141 / cdiv 8642 2c2 8983 sincsin 11665 cosccos 11666 Οcpi 11668 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-mulrcl 7923 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-precex 7934 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 ax-pre-mulgt0 7941 ax-pre-mulext 7942 ax-arch 7943 ax-caucvg 7944 ax-pre-suploc 7945 ax-addf 7946 ax-mulf 7947 |
This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-disj 3993 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-po 4308 df-iso 4309 df-iord 4378 df-on 4380 df-ilim 4381 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-isom 5237 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-of 6096 df-1st 6154 df-2nd 6155 df-recs 6319 df-irdg 6384 df-frec 6405 df-1o 6430 df-oadd 6434 df-er 6548 df-map 6663 df-pm 6664 df-en 6754 df-dom 6755 df-fin 6756 df-sup 6996 df-inf 6997 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-reap 8545 df-ap 8552 df-div 8643 df-inn 8933 df-2 8991 df-3 8992 df-4 8993 df-5 8994 df-6 8995 df-7 8996 df-8 8997 df-9 8998 df-n0 9190 df-z 9267 df-uz 9542 df-q 9633 df-rp 9667 df-xneg 9785 df-xadd 9786 df-ioo 9905 df-ioc 9906 df-ico 9907 df-icc 9908 df-fz 10022 df-fzo 10156 df-seqfrec 10459 df-exp 10533 df-fac 10719 df-bc 10741 df-ihash 10769 df-shft 10837 df-cj 10864 df-re 10865 df-im 10866 df-rsqrt 11020 df-abs 11021 df-clim 11300 df-sumdc 11375 df-ef 11669 df-sin 11671 df-cos 11672 df-pi 11674 df-rest 12707 df-topgen 12726 df-psmet 13704 df-xmet 13705 df-met 13706 df-bl 13707 df-mopn 13708 df-top 13769 df-topon 13782 df-bases 13814 df-ntr 13867 df-cn 13959 df-cnp 13960 df-tx 14024 df-cncf 14329 df-limced 14396 df-dvap 14397 |
This theorem is referenced by: sincos4thpi 14532 sincos6thpi 14534 |
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