Article SIN_COS6, MML version 4.99.1005

:: SIN_COS6:th 1
theorem
for b1, b2 being real set
      st 0 <= b1 & b1 < b2
   holds [\b1 / b2/] = 0;

:: SIN_COS6:th 2
theorem
for b1 being Relation-like Function-like set
for b2, b3 being set
      st b1 | b2 is one-to-one & b3 c= b2
   holds b1 | b3 is one-to-one;

:: SIN_COS6:th 3
theorem
for b1 being real set holds
   - 1 <= sin b1;

:: SIN_COS6:th 4
theorem
for b1 being real set holds
   sin b1 <= 1;

:: SIN_COS6:th 5
theorem
for b1 being real set holds
   - 1 <= cos b1;

:: SIN_COS6:th 6
theorem
for b1 being real set holds
   cos b1 <= 1;

:: SIN_COS6:funcreg 1
registration
  cluster PI -> real positive;
end;

:: SIN_COS6:th 7
theorem
sin - (PI / 2) = - 1 &
 sin . - (PI / 2) = - 1;

:: SIN_COS6:th 8
theorem
for b1 being real set
for b2 being integer set holds
   sin . b1 = sin . (b1 + ((2 * PI) * b2));

:: SIN_COS6:th 9
theorem
cos - (PI / 2) = 0 &
 cos . - (PI / 2) = 0;

:: SIN_COS6:th 10
theorem
for b1 being real set
for b2 being integer set holds
   cos . b1 = cos . (b1 + ((2 * PI) * b2));

:: SIN_COS6:th 11
theorem
for b1 being real set
for b2 being integer set
      st (2 * PI) * b2 < b1 &
         b1 < PI + ((2 * PI) * b2)
   holds 0 < sin b1;

:: SIN_COS6:th 12
theorem
for b1 being real set
for b2 being integer set
      st PI + ((2 * PI) * b2) < b1 &
         b1 < (2 * PI) + ((2 * PI) * b2)
   holds sin b1 < 0;

:: SIN_COS6:th 13
theorem
for b1 being real set
for b2 being integer set
      st (- (PI / 2)) + ((2 * PI) * b2) < b1 &
         b1 < (PI / 2) + ((2 * PI) * b2)
   holds 0 < cos b1;

:: SIN_COS6:th 14
theorem
for b1 being real set
for b2 being integer set
      st (PI / 2) + ((2 * PI) * b2) < b1 &
         b1 < ((3 / 2) * PI) + ((2 * PI) * b2)
   holds cos b1 < 0;

:: SIN_COS6:th 15
theorem
for b1 being real set
for b2 being integer set
      st ((3 / 2) * PI) + ((2 * PI) * b2) < b1 &
         b1 < (2 * PI) + ((2 * PI) * b2)
   holds 0 < cos b1;

:: SIN_COS6:th 16
theorem
for b1 being real set
for b2 being integer set
      st (2 * PI) * b2 <= b1 &
         b1 <= PI + ((2 * PI) * b2)
   holds 0 <= sin b1;

:: SIN_COS6:th 17
theorem
for b1 being real set
for b2 being integer set
      st PI + ((2 * PI) * b2) <= b1 &
         b1 <= (2 * PI) + ((2 * PI) * b2)
   holds sin b1 <= 0;

:: SIN_COS6:th 18
theorem
for b1 being real set
for b2 being integer set
      st (- (PI / 2)) + ((2 * PI) * b2) <= b1 &
         b1 <= (PI / 2) + ((2 * PI) * b2)
   holds 0 <= cos b1;

:: SIN_COS6:th 19
theorem
for b1 being real set
for b2 being integer set
      st (PI / 2) + ((2 * PI) * b2) <= b1 &
         b1 <= ((3 / 2) * PI) + ((2 * PI) * b2)
   holds cos b1 <= 0;

:: SIN_COS6:th 20
theorem
for b1 being real set
for b2 being integer set
      st ((3 / 2) * PI) + ((2 * PI) * b2) <= b1 &
         b1 <= (2 * PI) + ((2 * PI) * b2)
   holds 0 <= cos b1;

:: SIN_COS6:th 21
theorem
for b1 being real set
for b2 being integer set
      st (2 * PI) * b2 <= b1 &
         b1 < (2 * PI) + ((2 * PI) * b2) &
         sin b1 = 0 &
         b1 <> (2 * PI) * b2
   holds b1 = PI + ((2 * PI) * b2);

:: SIN_COS6:th 22
theorem
for b1 being real set
for b2 being integer set
      st (2 * PI) * b2 <= b1 &
         b1 < (2 * PI) + ((2 * PI) * b2) &
         cos b1 = 0 &
         b1 <> (PI / 2) + ((2 * PI) * b2)
   holds b1 = ((3 / 2) * PI) + ((2 * PI) * b2);

:: SIN_COS6:th 23
theorem
for b1 being real set
      st sin b1 = - 1
   holds b1 = ((3 / 2) * PI) + ((2 * PI) * [\b1 / (2 * PI)/]);

:: SIN_COS6:th 24
theorem
for b1 being real set
      st sin b1 = 1
   holds b1 = (PI / 2) + ((2 * PI) * [\b1 / (2 * PI)/]);

:: SIN_COS6:th 25
theorem
for b1 being real set
      st cos b1 = - 1
   holds b1 = PI + ((2 * PI) * [\b1 / (2 * PI)/]);

:: SIN_COS6:th 26
theorem
for b1 being real set
      st cos b1 = 1
   holds b1 = (2 * PI) * [\b1 / (2 * PI)/];

:: SIN_COS6:th 27
theorem
for b1 being real set
      st 0 <= b1 & b1 <= 2 * PI & sin b1 = - 1
   holds b1 = (3 / 2) * PI;

:: SIN_COS6:th 28
theorem
for b1 being real set
      st 0 <= b1 & b1 <= 2 * PI & sin b1 = 1
   holds b1 = PI / 2;

:: SIN_COS6:th 29
theorem
for b1 being real set
      st 0 <= b1 & b1 <= 2 * PI & cos b1 = - 1
   holds b1 = PI;

:: SIN_COS6:th 30
theorem
for b1 being real set
      st 0 <= b1 & b1 < PI / 2
   holds sin b1 < 1;

:: SIN_COS6:th 31
theorem
for b1 being real set
      st 0 <= b1 & b1 < (3 / 2) * PI
   holds - 1 < sin b1;

:: SIN_COS6:th 32
theorem
for b1 being real set
      st (3 / 2) * PI < b1 & b1 <= 2 * PI
   holds - 1 < sin b1;

:: SIN_COS6:th 33
theorem
for b1 being real set
      st PI / 2 < b1 & b1 <= 2 * PI
   holds sin b1 < 1;

:: SIN_COS6:th 34
theorem
for b1 being real set
      st 0 < b1 & b1 < 2 * PI
   holds cos b1 < 1;

:: SIN_COS6:th 35
theorem
for b1 being real set
      st 0 <= b1 & b1 < PI
   holds - 1 < cos b1;

:: SIN_COS6:th 36
theorem
for b1 being real set
      st PI < b1 & b1 <= 2 * PI
   holds - 1 < cos b1;

:: SIN_COS6:th 37
theorem
for b1 being real set
for b2 being integer set
      st (2 * PI) * b2 <= b1 &
         b1 < (PI / 2) + ((2 * PI) * b2)
   holds sin b1 < 1;

:: SIN_COS6:th 38
theorem
for b1 being real set
for b2 being integer set
      st (2 * PI) * b2 <= b1 &
         b1 < ((3 / 2) * PI) + ((2 * PI) * b2)
   holds - 1 < sin b1;

:: SIN_COS6:th 39
theorem
for b1 being real set
for b2 being integer set
      st ((3 / 2) * PI) + ((2 * PI) * b2) < b1 &
         b1 <= (2 * PI) + ((2 * PI) * b2)
   holds - 1 < sin b1;

:: SIN_COS6:th 40
theorem
for b1 being real set
for b2 being integer set
      st (PI / 2) + ((2 * PI) * b2) < b1 &
         b1 <= (2 * PI) + ((2 * PI) * b2)
   holds sin b1 < 1;

:: SIN_COS6:th 41
theorem
for b1 being real set
for b2 being integer set
      st (2 * PI) * b2 < b1 &
         b1 < (2 * PI) + ((2 * PI) * b2)
   holds cos b1 < 1;

:: SIN_COS6:th 42
theorem
for b1 being real set
for b2 being integer set
      st (2 * PI) * b2 <= b1 &
         b1 < PI + ((2 * PI) * b2)
   holds - 1 < cos b1;

:: SIN_COS6:th 43
theorem
for b1 being real set
for b2 being integer set
      st PI + ((2 * PI) * b2) < b1 &
         b1 <= (2 * PI) + ((2 * PI) * b2)
   holds - 1 < cos b1;

:: SIN_COS6:th 44
theorem
for b1 being real set
      st cos ((2 * PI) * b1) = 1
   holds b1 in INT;

:: SIN_COS6:th 45
theorem
sin .: [.- (PI / 2),PI / 2.] = [.- 1,1.];

:: SIN_COS6:th 46
theorem
sin .: ].- (PI / 2),PI / 2.[ = ].- 1,1.[;

:: SIN_COS6:th 47
theorem
sin .: [.PI / 2,(3 / 2) * PI.] = [.- 1,1.];

:: SIN_COS6:th 48
theorem
sin .: ].PI / 2,(3 / 2) * PI.[ = ].- 1,1.[;

:: SIN_COS6:th 49
theorem
cos .: [.0,PI.] = [.- 1,1.];

:: SIN_COS6:th 50
theorem
cos .: ].0,PI.[ = ].- 1,1.[;

:: SIN_COS6:th 51
theorem
cos .: [.PI,2 * PI.] = [.- 1,1.];

:: SIN_COS6:th 52
theorem
cos .: ].PI,2 * PI.[ = ].- 1,1.[;

:: SIN_COS6:th 53
theorem
for b1 being integer set holds
   sin is_increasing_on [.(- (PI / 2)) + ((2 * PI) * b1),(PI / 2) + ((2 * PI) * b1).];

:: SIN_COS6:th 54
theorem
for b1 being integer set holds
   sin is_decreasing_on [.(PI / 2) + ((2 * PI) * b1),((3 / 2) * PI) + ((2 * PI) * b1).];

:: SIN_COS6:th 55
theorem
for b1 being integer set holds
   cos is_decreasing_on [.(2 * PI) * b1,PI + ((2 * PI) * b1).];

:: SIN_COS6:th 56
theorem
for b1 being integer set holds
   cos is_increasing_on [.PI + ((2 * PI) * b1),(2 * PI) + ((2 * PI) * b1).];

:: SIN_COS6:th 57
theorem
for b1 being integer set holds
   sin | [.(- (PI / 2)) + ((2 * PI) * b1),(PI / 2) + ((2 * PI) * b1).] is one-to-one;

:: SIN_COS6:th 58
theorem
for b1 being integer set holds
   sin | [.(PI / 2) + ((2 * PI) * b1),((3 / 2) * PI) + ((2 * PI) * b1).] is one-to-one;

:: SIN_COS6:funcreg 2
registration
  cluster sin | [.- (PI / 2),PI / 2.] -> Relation-like one-to-one;
end;

:: SIN_COS6:funcreg 3
registration
  cluster sin | [.PI / 2,(3 / 2) * PI.] -> Relation-like one-to-one;
end;

:: SIN_COS6:funcreg 4
registration
  cluster sin | [.- (PI / 2),0.] -> Relation-like one-to-one;
end;

:: SIN_COS6:funcreg 5
registration
  cluster sin | [.0,PI / 2.] -> Relation-like one-to-one;
end;

:: SIN_COS6:funcreg 6
registration
  cluster sin | [.PI / 2,PI.] -> Relation-like one-to-one;
end;

:: SIN_COS6:funcreg 7
registration
  cluster sin | [.PI,(3 / 2) * PI.] -> Relation-like one-to-one;
end;

:: SIN_COS6:funcreg 8
registration
  cluster sin | [.(3 / 2) * PI,2 * PI.] -> Relation-like one-to-one;
end;

:: SIN_COS6:funcreg 9
registration
  cluster sin | ].- (PI / 2),PI / 2.[ -> Relation-like one-to-one;
end;

:: SIN_COS6:funcreg 10
registration
  cluster sin | ].PI / 2,(3 / 2) * PI.[ -> Relation-like one-to-one;
end;

:: SIN_COS6:funcreg 11
registration
  cluster sin | ].- (PI / 2),0.[ -> Relation-like one-to-one;
end;

:: SIN_COS6:funcreg 12
registration
  cluster sin | ].0,PI / 2.[ -> Relation-like one-to-one;
end;

:: SIN_COS6:funcreg 13
registration
  cluster sin | ].PI / 2,PI.[ -> Relation-like one-to-one;
end;

:: SIN_COS6:funcreg 14
registration
  cluster sin | ].PI,(3 / 2) * PI.[ -> Relation-like one-to-one;
end;

:: SIN_COS6:funcreg 15
registration
  cluster sin | ].(3 / 2) * PI,2 * PI.[ -> Relation-like one-to-one;
end;

:: SIN_COS6:th 59
theorem
for b1 being integer set holds
   cos | [.(2 * PI) * b1,PI + ((2 * PI) * b1).] is one-to-one;

:: SIN_COS6:th 60
theorem
for b1 being integer set holds
   cos | [.PI + ((2 * PI) * b1),(2 * PI) + ((2 * PI) * b1).] is one-to-one;

:: SIN_COS6:funcreg 16
registration
  cluster cos | [.0,PI.] -> Relation-like one-to-one;
end;

:: SIN_COS6:funcreg 17
registration
  cluster cos | [.PI,2 * PI.] -> Relation-like one-to-one;
end;

:: SIN_COS6:funcreg 18
registration
  cluster cos | [.0,PI / 2.] -> Relation-like one-to-one;
end;

:: SIN_COS6:funcreg 19
registration
  cluster cos | [.PI / 2,PI.] -> Relation-like one-to-one;
end;

:: SIN_COS6:funcreg 20
registration
  cluster cos | [.PI,(3 / 2) * PI.] -> Relation-like one-to-one;
end;

:: SIN_COS6:funcreg 21
registration
  cluster cos | [.(3 / 2) * PI,2 * PI.] -> Relation-like one-to-one;
end;

:: SIN_COS6:funcreg 22
registration
  cluster cos | ].0,PI.[ -> Relation-like one-to-one;
end;

:: SIN_COS6:funcreg 23
registration
  cluster cos | ].PI,2 * PI.[ -> Relation-like one-to-one;
end;

:: SIN_COS6:funcreg 24
registration
  cluster cos | ].0,PI / 2.[ -> Relation-like one-to-one;
end;

:: SIN_COS6:funcreg 25
registration
  cluster cos | ].PI / 2,PI.[ -> Relation-like one-to-one;
end;

:: SIN_COS6:funcreg 26
registration
  cluster cos | ].PI,(3 / 2) * PI.[ -> Relation-like one-to-one;
end;

:: SIN_COS6:funcreg 27
registration
  cluster cos | ].(3 / 2) * PI,2 * PI.[ -> Relation-like one-to-one;
end;

:: SIN_COS6:th 61
theorem
for b1, b2 being real set
for b3 being integer set
      st (2 * PI) * b3 <= b1 &
         b1 < (2 * PI) + ((2 * PI) * b3) &
         (2 * PI) * b3 <= b2 &
         b2 < (2 * PI) + ((2 * PI) * b3) &
         sin b1 = sin b2 &
         cos b1 = cos b2
   holds b1 = b2;

:: SIN_COS6:funcnot 1 => SIN_COS6:func 1
definition
  func arcsin -> Function-like Relation of REAL,REAL equals
    (sin | [.- (PI / 2),PI / 2.]) ";
end;

:: SIN_COS6:def 1
theorem
arcsin = (sin | [.- (PI / 2),PI / 2.]) ";

:: SIN_COS6:funcnot 2 => SIN_COS6:func 2
definition
  let a1 be set;
  func arcsin A1 -> set equals
    arcsin . a1;
end;

:: SIN_COS6:def 2
theorem
for b1 being set holds
   arcsin b1 = arcsin . b1;

:: SIN_COS6:funcnot 3 => SIN_COS6:func 3
definition
  let a1 be set;
  redefine func arcsin a1 -> Element of REAL;
end;

:: SIN_COS6:th 63
theorem
rng arcsin = [.- (PI / 2),PI / 2.];

:: SIN_COS6:funcreg 28
registration
  cluster arcsin -> Function-like one-to-one;
end;

:: SIN_COS6:th 64
theorem
dom arcsin = [.- 1,1.];

:: SIN_COS6:th 65
theorem
arcsin * (sin | [.- (PI / 2),PI / 2.]) = id [.- 1,1.];

:: SIN_COS6:th 66
theorem
arcsin * (sin | [.- (PI / 2),PI / 2.]) = id [.- 1,1.];

:: SIN_COS6:th 67
theorem
(sin | [.- (PI / 2),PI / 2.]) * arcsin = id [.- (PI / 2),PI / 2.];

:: SIN_COS6:th 68
theorem
(sin | [.- (PI / 2),PI / 2.]) * arcsin = id [.- (PI / 2),PI / 2.];

:: SIN_COS6:th 69
theorem
for b1 being real set
      st - 1 <= b1 & b1 <= 1
   holds sin arcsin b1 = b1;

:: SIN_COS6:th 70
theorem
for b1 being real set
      st - (PI / 2) <= b1 & b1 <= PI / 2
   holds arcsin sin b1 = b1;

:: SIN_COS6:th 71
theorem
arcsin - 1 = - (PI / 2);

:: SIN_COS6:th 72
theorem
arcsin 0 = 0;

:: SIN_COS6:th 73
theorem
arcsin 1 = PI / 2;

:: SIN_COS6:th 74
theorem
for b1 being real set
      st - 1 <= b1 & b1 <= 1 & arcsin b1 = - (PI / 2)
   holds b1 = - 1;

:: SIN_COS6:th 75
theorem
for b1 being real set
      st - 1 <= b1 & b1 <= 1 & arcsin b1 = 0
   holds b1 = 0;

:: SIN_COS6:th 76
theorem
for b1 being real set
      st - 1 <= b1 & b1 <= 1 & arcsin b1 = PI / 2
   holds b1 = 1;

:: SIN_COS6:th 77
theorem
for b1 being real set
      st - 1 <= b1 & b1 <= 1
   holds - (PI / 2) <= arcsin b1 & arcsin b1 <= PI / 2;

:: SIN_COS6:th 78
theorem
for b1 being real set
      st - 1 < b1 & b1 < 1
   holds - (PI / 2) < arcsin b1 & arcsin b1 < PI / 2;

:: SIN_COS6:th 79
theorem
for b1 being real set
      st - 1 <= b1 & b1 <= 1
   holds arcsin b1 = - arcsin - b1;

:: SIN_COS6:th 80
theorem
for b1, b2 being real set
      st 0 <= b1 & b2 ^2 + (b1 ^2) = 1
   holds cos arcsin b2 = b1;

:: SIN_COS6:th 81
theorem
for b1, b2 being real set
      st b1 <= 0 & b2 ^2 + (b1 ^2) = 1
   holds cos arcsin b2 = - b1;

:: SIN_COS6:th 82
theorem
for b1 being real set
      st - 1 <= b1 & b1 <= 1
   holds cos arcsin b1 = sqrt (1 - (b1 ^2));

:: SIN_COS6:th 83
theorem
arcsin is_increasing_on [.- 1,1.];

:: SIN_COS6:th 84
theorem
for b1 being real set holds
   arcsin is_differentiable_on ].- 1,1.[ &
    (- 1 < b1 & b1 < 1 implies diff(arcsin,b1) = 1 / sqrt (1 - (b1 ^2)));

:: SIN_COS6:th 85
theorem
arcsin is_continuous_on [.- 1,1.];

:: SIN_COS6:funcnot 4 => SIN_COS6:func 4
definition
  func arccos -> Function-like Relation of REAL,REAL equals
    (cos | [.0,PI.]) ";
end;

:: SIN_COS6:def 3
theorem
arccos = (cos | [.0,PI.]) ";

:: SIN_COS6:funcnot 5 => SIN_COS6:func 5
definition
  let a1 be set;
  func arccos A1 -> set equals
    arccos . a1;
end;

:: SIN_COS6:def 4
theorem
for b1 being set holds
   arccos b1 = arccos . b1;

:: SIN_COS6:funcnot 6 => SIN_COS6:func 6
definition
  let a1 be set;
  redefine func arccos a1 -> Element of REAL;
end;

:: SIN_COS6:th 87
theorem
rng arccos = [.0,PI.];

:: SIN_COS6:funcreg 29
registration
  cluster arccos -> Function-like one-to-one;
end;

:: SIN_COS6:th 88
theorem
dom arccos = [.- 1,1.];

:: SIN_COS6:th 89
theorem
arccos * (cos | [.0,PI.]) = id [.- 1,1.];

:: SIN_COS6:th 90
theorem
arccos * (cos | [.0,PI.]) = id [.- 1,1.];

:: SIN_COS6:th 91
theorem
(cos | [.0,PI.]) * arccos = id [.0,PI.];

:: SIN_COS6:th 92
theorem
(cos | [.0,PI.]) * arccos = id [.0,PI.];

:: SIN_COS6:th 93
theorem
for b1 being real set
      st - 1 <= b1 & b1 <= 1
   holds cos arccos b1 = b1;

:: SIN_COS6:th 94
theorem
for b1 being real set
      st 0 <= b1 & b1 <= PI
   holds arccos cos b1 = b1;

:: SIN_COS6:th 95
theorem
arccos - 1 = PI;

:: SIN_COS6:th 96
theorem
arccos 0 = PI / 2;

:: SIN_COS6:th 97
theorem
arccos 1 = 0;

:: SIN_COS6:th 98
theorem
for b1 being real set
      st - 1 <= b1 & b1 <= 1 & arccos b1 = 0
   holds b1 = 1;

:: SIN_COS6:th 99
theorem
for b1 being real set
      st - 1 <= b1 & b1 <= 1 & arccos b1 = PI / 2
   holds b1 = 0;

:: SIN_COS6:th 100
theorem
for b1 being real set
      st - 1 <= b1 & b1 <= 1 & arccos b1 = PI
   holds b1 = - 1;

:: SIN_COS6:th 101
theorem
for b1 being real set
      st - 1 <= b1 & b1 <= 1
   holds 0 <= arccos b1 & arccos b1 <= PI;

:: SIN_COS6:th 102
theorem
for b1 being real set
      st - 1 < b1 & b1 < 1
   holds 0 < arccos b1 & arccos b1 < PI;

:: SIN_COS6:th 103
theorem
for b1 being real set
      st - 1 <= b1 & b1 <= 1
   holds arccos b1 = PI - arccos - b1;

:: SIN_COS6:th 104
theorem
for b1, b2 being real set
      st 0 <= b1 & b2 ^2 + (b1 ^2) = 1
   holds sin arccos b2 = b1;

:: SIN_COS6:th 105
theorem
for b1, b2 being real set
      st b1 <= 0 & b2 ^2 + (b1 ^2) = 1
   holds sin arccos b2 = - b1;

:: SIN_COS6:th 106
theorem
for b1 being real set
      st - 1 <= b1 & b1 <= 1
   holds sin arccos b1 = sqrt (1 - (b1 ^2));

:: SIN_COS6:th 107
theorem
arccos is_decreasing_on [.- 1,1.];

:: SIN_COS6:th 108
theorem
for b1 being real set holds
   arccos is_differentiable_on ].- 1,1.[ &
    (- 1 < b1 & b1 < 1 implies diff(arccos,b1) = - (1 / sqrt (1 - (b1 ^2))));

:: SIN_COS6:th 109
theorem
arccos is_continuous_on [.- 1,1.];

:: SIN_COS6:th 110
theorem
for b1 being real set
      st - 1 <= b1 & b1 <= 1
   holds (arcsin b1) + arccos b1 = PI / 2;

:: SIN_COS6:th 111
theorem
for b1 being real set
      st - 1 <= b1 & b1 <= 1
   holds (arccos - b1) - arcsin b1 = PI / 2;

:: SIN_COS6:th 112
theorem
for b1 being real set
      st - 1 <= b1 & b1 <= 1
   holds (arccos b1) - arcsin - b1 = PI / 2;