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Theorem sincosq3sgn 12925
Description: The signs of the sine and cosine functions in the third quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)
Assertion
Ref Expression
sincosq3sgn  |-  ( A  e.  ( pi (,) ( 3  x.  (
pi  /  2 ) ) )  ->  (
( sin `  A
)  <  0  /\  ( cos `  A )  <  0 ) )

Proof of Theorem sincosq3sgn
StepHypRef Expression
1 pire 12883 . . 3  |-  pi  e.  RR
2 3re 8801 . . . 4  |-  3  e.  RR
3 halfpire 12889 . . . 4  |-  ( pi 
/  2 )  e.  RR
42, 3remulcli 7787 . . 3  |-  ( 3  x.  ( pi  / 
2 ) )  e.  RR
5 rexr 7818 . . . 4  |-  ( pi  e.  RR  ->  pi  e.  RR* )
6 rexr 7818 . . . 4  |-  ( ( 3  x.  ( pi 
/  2 ) )  e.  RR  ->  (
3  x.  ( pi 
/  2 ) )  e.  RR* )
7 elioo2 9711 . . . 4  |-  ( ( pi  e.  RR*  /\  (
3  x.  ( pi 
/  2 ) )  e.  RR* )  ->  ( A  e.  ( pi (,) ( 3  x.  (
pi  /  2 ) ) )  <->  ( A  e.  RR  /\  pi  <  A  /\  A  <  (
3  x.  ( pi 
/  2 ) ) ) ) )
85, 6, 7syl2an 287 . . 3  |-  ( ( pi  e.  RR  /\  ( 3  x.  (
pi  /  2 ) )  e.  RR )  ->  ( A  e.  ( pi (,) (
3  x.  ( pi 
/  2 ) ) )  <->  ( A  e.  RR  /\  pi  <  A  /\  A  <  (
3  x.  ( pi 
/  2 ) ) ) ) )
91, 4, 8mp2an 422 . 2  |-  ( A  e.  ( pi (,) ( 3  x.  (
pi  /  2 ) ) )  <->  ( A  e.  RR  /\  pi  <  A  /\  A  <  (
3  x.  ( pi 
/  2 ) ) ) )
10 pidiv2halves 12892 . . . . . . . . 9  |-  ( ( pi  /  2 )  +  ( pi  / 
2 ) )  =  pi
1110breq1i 3936 . . . . . . . 8  |-  ( ( ( pi  /  2
)  +  ( pi 
/  2 ) )  <  A  <->  pi  <  A )
12 ltaddsub 8205 . . . . . . . . 9  |-  ( ( ( pi  /  2
)  e.  RR  /\  ( pi  /  2
)  e.  RR  /\  A  e.  RR )  ->  ( ( ( pi 
/  2 )  +  ( pi  /  2
) )  <  A  <->  ( pi  /  2 )  <  ( A  -  ( pi  /  2
) ) ) )
133, 3, 12mp3an12 1305 . . . . . . . 8  |-  ( A  e.  RR  ->  (
( ( pi  / 
2 )  +  ( pi  /  2 ) )  <  A  <->  ( pi  /  2 )  <  ( A  -  ( pi  /  2 ) ) ) )
1411, 13bitr3id 193 . . . . . . 7  |-  ( A  e.  RR  ->  (
pi  <  A  <->  ( pi  /  2 )  <  ( A  -  ( pi  /  2 ) ) ) )
15 ltsubadd 8201 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( pi  /  2
)  e.  RR  /\  pi  e.  RR )  -> 
( ( A  -  ( pi  /  2
) )  <  pi  <->  A  <  ( pi  +  ( pi  /  2
) ) ) )
163, 1, 15mp3an23 1307 . . . . . . . 8  |-  ( A  e.  RR  ->  (
( A  -  (
pi  /  2 ) )  <  pi  <->  A  <  ( pi  +  ( pi 
/  2 ) ) ) )
17 df-3 8787 . . . . . . . . . . 11  |-  3  =  ( 2  +  1 )
1817oveq1i 5784 . . . . . . . . . 10  |-  ( 3  x.  ( pi  / 
2 ) )  =  ( ( 2  +  1 )  x.  (
pi  /  2 ) )
19 2cn 8798 . . . . . . . . . . 11  |-  2  e.  CC
20 ax-1cn 7720 . . . . . . . . . . 11  |-  1  e.  CC
213recni 7785 . . . . . . . . . . 11  |-  ( pi 
/  2 )  e.  CC
2219, 20, 21adddiri 7784 . . . . . . . . . 10  |-  ( ( 2  +  1 )  x.  ( pi  / 
2 ) )  =  ( ( 2  x.  ( pi  /  2
) )  +  ( 1  x.  ( pi 
/  2 ) ) )
231recni 7785 . . . . . . . . . . . 12  |-  pi  e.  CC
24 2ap0 8820 . . . . . . . . . . . 12  |-  2 #  0
2523, 19, 24divcanap2i 8522 . . . . . . . . . . 11  |-  ( 2  x.  ( pi  / 
2 ) )  =  pi
2621mulid2i 7776 . . . . . . . . . . 11  |-  ( 1  x.  ( pi  / 
2 ) )  =  ( pi  /  2
)
2725, 26oveq12i 5786 . . . . . . . . . 10  |-  ( ( 2  x.  ( pi 
/  2 ) )  +  ( 1  x.  ( pi  /  2
) ) )  =  ( pi  +  ( pi  /  2 ) )
2818, 22, 273eqtrri 2165 . . . . . . . . 9  |-  ( pi  +  ( pi  / 
2 ) )  =  ( 3  x.  (
pi  /  2 ) )
2928breq2i 3937 . . . . . . . 8  |-  ( A  <  ( pi  +  ( pi  /  2
) )  <->  A  <  ( 3  x.  ( pi 
/  2 ) ) )
3016, 29syl6rbb 196 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  <  ( 3  x.  ( pi  /  2
) )  <->  ( A  -  ( pi  / 
2 ) )  < 
pi ) )
3114, 30anbi12d 464 . . . . . 6  |-  ( A  e.  RR  ->  (
( pi  <  A  /\  A  <  ( 3  x.  ( pi  / 
2 ) ) )  <-> 
( ( pi  / 
2 )  <  ( A  -  ( pi  /  2 ) )  /\  ( A  -  (
pi  /  2 ) )  <  pi ) ) )
32 resubcl 8033 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( pi  /  2
)  e.  RR )  ->  ( A  -  ( pi  /  2
) )  e.  RR )
333, 32mpan2 421 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  -  ( pi  /  2 ) )  e.  RR )
34 sincosq2sgn 12924 . . . . . . . . 9  |-  ( ( A  -  ( pi 
/  2 ) )  e.  ( ( pi 
/  2 ) (,) pi )  ->  (
0  <  ( sin `  ( A  -  (
pi  /  2 ) ) )  /\  ( cos `  ( A  -  ( pi  /  2
) ) )  <  0 ) )
35 rexr 7818 . . . . . . . . . . 11  |-  ( ( pi  /  2 )  e.  RR  ->  (
pi  /  2 )  e.  RR* )
36 elioo2 9711 . . . . . . . . . . 11  |-  ( ( ( pi  /  2
)  e.  RR*  /\  pi  e.  RR* )  ->  (
( A  -  (
pi  /  2 ) )  e.  ( ( pi  /  2 ) (,) pi )  <->  ( ( A  -  ( pi  /  2 ) )  e.  RR  /\  ( pi 
/  2 )  < 
( A  -  (
pi  /  2 ) )  /\  ( A  -  ( pi  / 
2 ) )  < 
pi ) ) )
3735, 5, 36syl2an 287 . . . . . . . . . 10  |-  ( ( ( pi  /  2
)  e.  RR  /\  pi  e.  RR )  -> 
( ( A  -  ( pi  /  2
) )  e.  ( ( pi  /  2
) (,) pi )  <-> 
( ( A  -  ( pi  /  2
) )  e.  RR  /\  ( pi  /  2
)  <  ( A  -  ( pi  / 
2 ) )  /\  ( A  -  (
pi  /  2 ) )  <  pi ) ) )
383, 1, 37mp2an 422 . . . . . . . . 9  |-  ( ( A  -  ( pi 
/  2 ) )  e.  ( ( pi 
/  2 ) (,) pi )  <->  ( ( A  -  ( pi  /  2 ) )  e.  RR  /\  ( pi 
/  2 )  < 
( A  -  (
pi  /  2 ) )  /\  ( A  -  ( pi  / 
2 ) )  < 
pi ) )
39 ancom 264 . . . . . . . . 9  |-  ( ( 0  <  ( sin `  ( A  -  (
pi  /  2 ) ) )  /\  ( cos `  ( A  -  ( pi  /  2
) ) )  <  0 )  <->  ( ( cos `  ( A  -  ( pi  /  2
) ) )  <  0  /\  0  < 
( sin `  ( A  -  ( pi  /  2 ) ) ) ) )
4034, 38, 393imtr3i 199 . . . . . . . 8  |-  ( ( ( A  -  (
pi  /  2 ) )  e.  RR  /\  ( pi  /  2
)  <  ( A  -  ( pi  / 
2 ) )  /\  ( A  -  (
pi  /  2 ) )  <  pi )  ->  ( ( cos `  ( A  -  (
pi  /  2 ) ) )  <  0  /\  0  <  ( sin `  ( A  -  (
pi  /  2 ) ) ) ) )
4133, 40syl3an1 1249 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( pi  /  2
)  <  ( A  -  ( pi  / 
2 ) )  /\  ( A  -  (
pi  /  2 ) )  <  pi )  ->  ( ( cos `  ( A  -  (
pi  /  2 ) ) )  <  0  /\  0  <  ( sin `  ( A  -  (
pi  /  2 ) ) ) ) )
42413expib 1184 . . . . . 6  |-  ( A  e.  RR  ->  (
( ( pi  / 
2 )  <  ( A  -  ( pi  /  2 ) )  /\  ( A  -  (
pi  /  2 ) )  <  pi )  ->  ( ( cos `  ( A  -  (
pi  /  2 ) ) )  <  0  /\  0  <  ( sin `  ( A  -  (
pi  /  2 ) ) ) ) ) )
4331, 42sylbid 149 . . . . 5  |-  ( A  e.  RR  ->  (
( pi  <  A  /\  A  <  ( 3  x.  ( pi  / 
2 ) ) )  ->  ( ( cos `  ( A  -  (
pi  /  2 ) ) )  <  0  /\  0  <  ( sin `  ( A  -  (
pi  /  2 ) ) ) ) ) )
4433resincld 11436 . . . . . . 7  |-  ( A  e.  RR  ->  ( sin `  ( A  -  ( pi  /  2
) ) )  e.  RR )
4544lt0neg2d 8285 . . . . . 6  |-  ( A  e.  RR  ->  (
0  <  ( sin `  ( A  -  (
pi  /  2 ) ) )  <->  -u ( sin `  ( A  -  (
pi  /  2 ) ) )  <  0
) )
4645anbi2d 459 . . . . 5  |-  ( A  e.  RR  ->  (
( ( cos `  ( A  -  ( pi  /  2 ) ) )  <  0  /\  0  <  ( sin `  ( A  -  ( pi  /  2 ) ) ) )  <->  ( ( cos `  ( A  -  (
pi  /  2 ) ) )  <  0  /\  -u ( sin `  ( A  -  ( pi  /  2 ) ) )  <  0 ) ) )
4743, 46sylibd 148 . . . 4  |-  ( A  e.  RR  ->  (
( pi  <  A  /\  A  <  ( 3  x.  ( pi  / 
2 ) ) )  ->  ( ( cos `  ( A  -  (
pi  /  2 ) ) )  <  0  /\  -u ( sin `  ( A  -  ( pi  /  2 ) ) )  <  0 ) ) )
48 recn 7760 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  CC )
49 pncan3 7977 . . . . . . . . 9  |-  ( ( ( pi  /  2
)  e.  CC  /\  A  e.  CC )  ->  ( ( pi  / 
2 )  +  ( A  -  ( pi 
/  2 ) ) )  =  A )
5021, 48, 49sylancr 410 . . . . . . . 8  |-  ( A  e.  RR  ->  (
( pi  /  2
)  +  ( A  -  ( pi  / 
2 ) ) )  =  A )
5150fveq2d 5425 . . . . . . 7  |-  ( A  e.  RR  ->  ( sin `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  =  ( sin `  A
) )
5233recnd 7801 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  -  ( pi  /  2 ) )  e.  CC )
53 sinhalfpip 12917 . . . . . . . 8  |-  ( ( A  -  ( pi 
/  2 ) )  e.  CC  ->  ( sin `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  =  ( cos `  ( A  -  ( pi  /  2 ) ) ) )
5452, 53syl 14 . . . . . . 7  |-  ( A  e.  RR  ->  ( sin `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  =  ( cos `  ( A  -  ( pi  /  2 ) ) ) )
5551, 54eqtr3d 2174 . . . . . 6  |-  ( A  e.  RR  ->  ( sin `  A )  =  ( cos `  ( A  -  ( pi  /  2 ) ) ) )
5655breq1d 3939 . . . . 5  |-  ( A  e.  RR  ->  (
( sin `  A
)  <  0  <->  ( cos `  ( A  -  (
pi  /  2 ) ) )  <  0
) )
5750fveq2d 5425 . . . . . . 7  |-  ( A  e.  RR  ->  ( cos `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  =  ( cos `  A
) )
58 coshalfpip 12919 . . . . . . . 8  |-  ( ( A  -  ( pi 
/  2 ) )  e.  CC  ->  ( cos `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  = 
-u ( sin `  ( A  -  ( pi  /  2 ) ) ) )
5952, 58syl 14 . . . . . . 7  |-  ( A  e.  RR  ->  ( cos `  ( ( pi 
/  2 )  +  ( A  -  (
pi  /  2 ) ) ) )  = 
-u ( sin `  ( A  -  ( pi  /  2 ) ) ) )
6057, 59eqtr3d 2174 . . . . . 6  |-  ( A  e.  RR  ->  ( cos `  A )  = 
-u ( sin `  ( A  -  ( pi  /  2 ) ) ) )
6160breq1d 3939 . . . . 5  |-  ( A  e.  RR  ->  (
( cos `  A
)  <  0  <->  -u ( sin `  ( A  -  (
pi  /  2 ) ) )  <  0
) )
6256, 61anbi12d 464 . . . 4  |-  ( A  e.  RR  ->  (
( ( sin `  A
)  <  0  /\  ( cos `  A )  <  0 )  <->  ( ( cos `  ( A  -  ( pi  /  2
) ) )  <  0  /\  -u ( sin `  ( A  -  ( pi  /  2
) ) )  <  0 ) ) )
6347, 62sylibrd 168 . . 3  |-  ( A  e.  RR  ->  (
( pi  <  A  /\  A  <  ( 3  x.  ( pi  / 
2 ) ) )  ->  ( ( sin `  A )  <  0  /\  ( cos `  A
)  <  0 ) ) )
64633impib 1179 . 2  |-  ( ( A  e.  RR  /\  pi  <  A  /\  A  <  ( 3  x.  (
pi  /  2 ) ) )  ->  (
( sin `  A
)  <  0  /\  ( cos `  A )  <  0 ) )
659, 64sylbi 120 1  |-  ( A  e.  ( pi (,) ( 3  x.  (
pi  /  2 ) ) )  ->  (
( sin `  A
)  <  0  /\  ( cos `  A )  <  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7625   RRcr 7626   0cc0 7627   1c1 7628    + caddc 7630    x. cmul 7632   RR*cxr 7806    < clt 7807    - cmin 7940   -ucneg 7941    / cdiv 8439   2c2 8778   3c3 8779   (,)cioo 9678   sincsin 11357   cosccos 11358   picpi 11360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745  ax-arch 7746  ax-caucvg 7747  ax-pre-suploc 7748  ax-addf 7749  ax-mulf 7750
This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-disj 3907  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-of 5982  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-map 6544  df-pm 6545  df-en 6635  df-dom 6636  df-fin 6637  df-sup 6871  df-inf 6872  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440  df-inn 8728  df-2 8786  df-3 8787  df-4 8788  df-5 8789  df-6 8790  df-7 8791  df-8 8792  df-9 8793  df-n0 8985  df-z 9062  df-uz 9334  df-q 9419  df-rp 9449  df-xneg 9566  df-xadd 9567  df-ioo 9682  df-ioc 9683  df-ico 9684  df-icc 9685  df-fz 9798  df-fzo 9927  df-seqfrec 10226  df-exp 10300  df-fac 10479  df-bc 10501  df-ihash 10529  df-shft 10594  df-cj 10621  df-re 10622  df-im 10623  df-rsqrt 10777  df-abs 10778  df-clim 11055  df-sumdc 11130  df-ef 11361  df-sin 11363  df-cos 11364  df-pi 11366  df-rest 12131  df-topgen 12150  df-psmet 12165  df-xmet 12166  df-met 12167  df-bl 12168  df-mopn 12169  df-top 12174  df-topon 12187  df-bases 12219  df-ntr 12274  df-cn 12366  df-cnp 12367  df-tx 12431  df-cncf 12736  df-limced 12803  df-dvap 12804
This theorem is referenced by:  sincosq4sgn  12926
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