Metamath Proof Explorer


Theorem sincosq3sgn

Description: The signs of the sine and cosine functions in the third quadrant. (Contributed by Paul Chapman, 24-Jan-2008)

Ref Expression
Assertion sincosq3sgn A π 3 π 2 sin A < 0 cos A < 0

Proof

Step Hyp Ref Expression
1 pire π
2 3re 3
3 halfpire π 2
4 2 3 remulcli 3 π 2
5 rexr π π *
6 rexr 3 π 2 3 π 2 *
7 elioo2 π * 3 π 2 * A π 3 π 2 A π < A A < 3 π 2
8 5 6 7 syl2an π 3 π 2 A π 3 π 2 A π < A A < 3 π 2
9 1 4 8 mp2an A π 3 π 2 A π < A A < 3 π 2
10 pidiv2halves π 2 + π 2 = π
11 10 breq1i π 2 + π 2 < A π < A
12 ltaddsub π 2 π 2 A π 2 + π 2 < A π 2 < A π 2
13 3 3 12 mp3an12 A π 2 + π 2 < A π 2 < A π 2
14 11 13 bitr3id A π < A π 2 < A π 2
15 ltsubadd A π 2 π A π 2 < π A < π + π 2
16 3 1 15 mp3an23 A A π 2 < π A < π + π 2
17 df-3 3 = 2 + 1
18 17 oveq1i 3 π 2 = 2 + 1 π 2
19 2cn 2
20 ax-1cn 1
21 3 recni π 2
22 19 20 21 adddiri 2 + 1 π 2 = 2 π 2 + 1 π 2
23 1 recni π
24 2ne0 2 0
25 23 19 24 divcan2i 2 π 2 = π
26 21 mulid2i 1 π 2 = π 2
27 25 26 oveq12i 2 π 2 + 1 π 2 = π + π 2
28 18 22 27 3eqtrri π + π 2 = 3 π 2
29 28 breq2i A < π + π 2 A < 3 π 2
30 16 29 bitr2di A A < 3 π 2 A π 2 < π
31 14 30 anbi12d A π < A A < 3 π 2 π 2 < A π 2 A π 2 < π
32 resubcl A π 2 A π 2
33 3 32 mpan2 A A π 2
34 sincosq2sgn A π 2 π 2 π 0 < sin A π 2 cos A π 2 < 0
35 rexr π 2 π 2 *
36 elioo2 π 2 * π * A π 2 π 2 π A π 2 π 2 < A π 2 A π 2 < π
37 35 5 36 syl2an π 2 π A π 2 π 2 π A π 2 π 2 < A π 2 A π 2 < π
38 3 1 37 mp2an A π 2 π 2 π A π 2 π 2 < A π 2 A π 2 < π
39 ancom 0 < sin A π 2 cos A π 2 < 0 cos A π 2 < 0 0 < sin A π 2
40 34 38 39 3imtr3i A π 2 π 2 < A π 2 A π 2 < π cos A π 2 < 0 0 < sin A π 2
41 33 40 syl3an1 A π 2 < A π 2 A π 2 < π cos A π 2 < 0 0 < sin A π 2
42 41 3expib A π 2 < A π 2 A π 2 < π cos A π 2 < 0 0 < sin A π 2
43 31 42 sylbid A π < A A < 3 π 2 cos A π 2 < 0 0 < sin A π 2
44 33 resincld A sin A π 2
45 44 lt0neg2d A 0 < sin A π 2 sin A π 2 < 0
46 45 anbi2d A cos A π 2 < 0 0 < sin A π 2 cos A π 2 < 0 sin A π 2 < 0
47 43 46 sylibd A π < A A < 3 π 2 cos A π 2 < 0 sin A π 2 < 0
48 recn A A
49 pncan3 π 2 A π 2 + A - π 2 = A
50 21 48 49 sylancr A π 2 + A - π 2 = A
51 50 fveq2d A sin π 2 + A - π 2 = sin A
52 33 recnd A A π 2
53 sinhalfpip A π 2 sin π 2 + A - π 2 = cos A π 2
54 52 53 syl A sin π 2 + A - π 2 = cos A π 2
55 51 54 eqtr3d A sin A = cos A π 2
56 55 breq1d A sin A < 0 cos A π 2 < 0
57 50 fveq2d A cos π 2 + A - π 2 = cos A
58 coshalfpip A π 2 cos π 2 + A - π 2 = sin A π 2
59 52 58 syl A cos π 2 + A - π 2 = sin A π 2
60 57 59 eqtr3d A cos A = sin A π 2
61 60 breq1d A cos A < 0 sin A π 2 < 0
62 56 61 anbi12d A sin A < 0 cos A < 0 cos A π 2 < 0 sin A π 2 < 0
63 47 62 sylibrd A π < A A < 3 π 2 sin A < 0 cos A < 0
64 63 3impib A π < A A < 3 π 2 sin A < 0 cos A < 0
65 9 64 sylbi A π 3 π 2 sin A < 0 cos A < 0