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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 14616 | . . . . . 6 β’ (Ο / 2) β β | |
2 | 1 | recni 7988 | . . . . 5 β’ (Ο / 2) β β |
3 | mulcl 7957 | . . . . 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 14647 | . . . 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 1020 | . 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 7967 | . . . . . . 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 1337 | . . . . . 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 1019 | . . . . 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 5898 | . . . . . . 7 β’ ((π΄ + π΅) = 1 β ((π΄ + π΅) Β· (Ο / 2)) = (1 Β· (Ο / 2))) | |
12 | 2 | mullidi 7979 | . . . . . . 7 β’ (1 Β· (Ο / 2)) = (Ο / 2) |
13 | 11, 12 | eqtrdi 2238 | . . . . . 6 β’ ((π΄ + π΅) = 1 β ((π΄ + π΅) Β· (Ο / 2)) = (Ο / 2)) |
14 | 13 | 3ad2ant3 1022 | . . . . 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 2224 | . . . 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 7957 | . . . . . . 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 8179 | . . . . . 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 1354 | . . . . 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 1019 | . . . 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 5534 | . 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 2224 | 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 980 = wceq 1364 β wcel 2160 βcfv 5231 (class class class)co 5891 βcc 7828 1c1 7831 + caddc 7833 Β· cmul 7835 β cmin 8147 / cdiv 8648 2c2 8989 sincsin 11671 cosccos 11672 Οcpi 11674 |
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 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7921 ax-resscn 7922 ax-1cn 7923 ax-1re 7924 ax-icn 7925 ax-addcl 7926 ax-addrcl 7927 ax-mulcl 7928 ax-mulrcl 7929 ax-addcom 7930 ax-mulcom 7931 ax-addass 7932 ax-mulass 7933 ax-distr 7934 ax-i2m1 7935 ax-0lt1 7936 ax-1rid 7937 ax-0id 7938 ax-rnegex 7939 ax-precex 7940 ax-cnre 7941 ax-pre-ltirr 7942 ax-pre-ltwlin 7943 ax-pre-lttrn 7944 ax-pre-apti 7945 ax-pre-ltadd 7946 ax-pre-mulgt0 7947 ax-pre-mulext 7948 ax-arch 7949 ax-caucvg 7950 ax-pre-suploc 7951 ax-addf 7952 ax-mulf 7953 |
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 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-disj 3996 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-po 4311 df-iso 4312 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-isom 5240 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-of 6101 df-1st 6159 df-2nd 6160 df-recs 6324 df-irdg 6389 df-frec 6410 df-1o 6435 df-oadd 6439 df-er 6553 df-map 6668 df-pm 6669 df-en 6759 df-dom 6760 df-fin 6761 df-sup 7002 df-inf 7003 df-pnf 8013 df-mnf 8014 df-xr 8015 df-ltxr 8016 df-le 8017 df-sub 8149 df-neg 8150 df-reap 8551 df-ap 8558 df-div 8649 df-inn 8939 df-2 8997 df-3 8998 df-4 8999 df-5 9000 df-6 9001 df-7 9002 df-8 9003 df-9 9004 df-n0 9196 df-z 9273 df-uz 9548 df-q 9639 df-rp 9673 df-xneg 9791 df-xadd 9792 df-ioo 9911 df-ioc 9912 df-ico 9913 df-icc 9914 df-fz 10028 df-fzo 10162 df-seqfrec 10465 df-exp 10539 df-fac 10725 df-bc 10747 df-ihash 10775 df-shft 10843 df-cj 10870 df-re 10871 df-im 10872 df-rsqrt 11026 df-abs 11027 df-clim 11306 df-sumdc 11381 df-ef 11675 df-sin 11677 df-cos 11678 df-pi 11680 df-rest 12718 df-topgen 12737 df-psmet 13823 df-xmet 13824 df-met 13825 df-bl 13826 df-mopn 13827 df-top 13901 df-topon 13914 df-bases 13946 df-ntr 13999 df-cn 14091 df-cnp 14092 df-tx 14156 df-cncf 14461 df-limced 14528 df-dvap 14529 |
This theorem is referenced by: sincos4thpi 14664 sincos6thpi 14666 |
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