tryg, Matematyka, fizyka, informatyka
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A
T
E
X
K
OMPENDIUM
T
RYGONOMETRYCZNE
BY
Q
UEZAK
W
ZORY
R
EDUKCYJNE
k at
45
◦
−
45
◦
+
90
◦
−
90
◦
+
180
◦
−
180
◦
+
270
◦
−
270
◦
+
360
◦
−
sin cos(45
◦
+
) cos(45
◦
−
) cos
cos
sin
−sin
−cos
−cos
−sin
cos sin(45
◦
+
) sin(45
◦
−
) sin
−sin
−cos
−cos
−sin
sin
cos
tg ctg(45
◦
+
) ctg(45
◦
−
) ctg
−ctg
−tg
tg
ctg
−ctg
−tg
ctg tg(45
◦
+
) tg(45
◦
−
)
tg
−tg
−ctg
ctg
tg
−tg
−ctg
W
YBRANE
W
ARTO SCI
k at
15
◦
22, 5
◦
30
◦
45
◦
60
◦
67, 5
◦
75
◦
90
◦
180
◦
270
◦
360
◦
√
6−
√
P
2−
√
√
√
3
2
P
√
2
√
√
2
2
2+
sin
2
1
2
2
6+
1
0 −1
0
P
P
4
2
2
2
4
√
√
√
2
√
3
2
√
2−
√
√
6−
√
2+
2
cos
6+
2
2
1
2
2
0 −1
0
1
tg 2−
P
4
P
2
√
3
3
2
P
P
2
4
P
3
2−1
1
3
2 + 1 2 +
3 × 0 × 0
P
P
P
√
3
3
P
2−1 2−
P
ctg 2 +
3
2 + 1
3 1
3 0 × 0 ×
sin
2
+ cos
2
= 1
tg
·ctg
= 1
SIN α
COS α
= tg
COS α
SIN α
= ctg
sin(
+
) = sin
cos
+ cos
sin
(1)
tg
+ tg
=
sin(
+
)
cos
cos
(24)
sin(
−
) = sin
cos
−cos
sin
(2)
tg
−tg
=
sin(
−
)
cos
cos
(25)
cos(
+
) = cos
cos
−sin
sin
(3)
ctg
+ ctg
=
sin(
+
)
sin
sin
cos(
−
) = cos
cos
+ sin
sin
(4)
(26)
tg
+ tg
1−tg
tg
tg(
+
) =
(5)
ctg
−ctg
=−
sin(
−
)
sin
sin
(27)
tg
−tg
1 + tg
tg
tg(
−
) =
(6)
1 + sin
= 2 sin
2
2
+ 45
◦
= 2 cos
2
2
−45
◦
(28)
ctg(
+
) =
ctg
ctg
−1
ctg
+ ctg
(7)
1−sin
= 2 sin
2
2
−45
◦
= 2 cos
2
2
+ 45
◦
(29)
1 + cos
= 2 cos
2
2
ctg(
−
) =
ctg
ctg
+ 1
ctg
−ctg
(8)
(30)
1−cos
= 2 sin
2
2
sin 2
= 2 sin
cos
(9)
cos 2
= 1−2 sin
2
= 2 cos
2
−1 = cos
2
−sin
2
(10)
(31)
r
sin
2
1−cos
2
=±
(⋆)
(32)
2tg
1−tg
2
tg2
=
(11)
r
cos
2
1 + cos
2
ctg2
=
ctg
2
−1
2ctg
=±
(⋆)
(33)
(12)
(⋆)
[znak zalezy od cwiartki, w której lezy ko ncowe ramie k ata
2
]
tg
2
sin 3
= 3 sin
−4 sin
3
(13)
=
1−cos
sin
sin
1 + cos
cos 3
= 4 cos
3
−3 cos
(14)
=
(34)
tg3
=
3tg
−tg
3
1−3tg
2
(15)
ctg
2
=
1 + cos
sin
=
sin
1−cos
(35)
P
P
ctg3
=
3ctg
−ctg
3
1−3ctg
2
2 sin(45
◦
+
) =
2 cos(45
◦
−
)
(36)
sin
+ cos
=
(16)
P
P
sin
−cos
=
2 sin(45
◦
−
)
(37)
sin
2
−sin
2
= sin(
+
) sin(
−
)
(38)
cos
2
−sin
2
= cos(
+
) cos(
−
)
(39)
cos
2
−cos
2
=−sin(
+
) sin(
−
)
(40)
2 cos(45
◦
+
) =
sin
cos
=
1
2
[sin(
+
) + sin(
−
)]
(17)
sin
sin
=
1
2
[cos(
−
)−cos(
+
)]
(18)
cos
cos
=
1
2
[cos(
+
) + cos(
−
)]
(19)
sin
=
2tg
2
1 + tg
2
2
(41)
sin
+ sin
= 2 sin
+
2
cos
−
2
(20)
cos
=
1−tg
2
2
1 + tg
2
2
(42)
sin
−sin
= 2 sin
−
2
cos
+
2
(21)
2tg
2
1−tg
2
2
tg
=
(43)
cos
+ cos
= 2 cos
+
2
cos
−
2
(22)
ctg
=
1−tg
2
2
2tg
2
=
ctg
2
2
−1
2ctg
2
cos
−cos
=−2 sin
+
2
sin
−
2
(23)
(44)
1
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