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اﻟﺪوال اﻟﻤﺜﻠﺜﻴﺔ
١{١
ﻗﻴﺎﺳﺎت اﻟﺰواﻳﺎ
١{١{١
Trigonometric Functions
EאאF 1 ( 0, 0 ) אאא
אאאאאאאאאא
θ
אאא
+ אאא
− אאא
אא
1
1 Wא
360
90
−45
270
E π ≈ 3.14 F 2π אאאK אאאאWאא
א 2π = 360
א π = 180
π
2
−
×π
⎯⎯180
⎯⎯
→ א
π
4
3π
2
×180
π →
א ⎯⎯⎯⎯
W
א
٢
اﻟﺪوال اﻟﻤﺜﻠﺜﻴﺔ
אאא
Wאאא θ Kאאא θ ∈
y
1
( x, y ) = ( cos θ,sin θ )
θ
−1
1
x
−1
(
(
א
) א א sin θ = y
θ
) א cos θ = x
cos θ
sin θ
K θ =
0
1
0
π
6
3
2
1
2
π
4
2
2
2
2
π
3
1
2
3
2
π
2
0
1
7π
cos θ ، sin θ א ١
6
אא θ Kא
y
7π
6
π
6
( cos 76π , sin 76π )
(−
3 1
,
2 2
1
x
)
K cos θ = −
1
3
، sin θ =
2
2
٣
→
sin :
Dsin =
,
,
Rsin = [ −1,1 ]
Dcos =
→
,
Rcos = [ −1,1 ]
,
1
1
1
1
(
א
) א
(
cos θ
sin θ
= − { 0, ±π, ±2π, …}
Dcot
) א א
sin θ
cos θ
π
3π
5π
= − { ± 2 , ± 2 , ± 2 , …}
tan θ =
cot θ =
Dtan
Rcot =
Rtan =
8
8
4
4
4
4
8
8
(
א
) א
(
1
sin θ
π
3π
5π
= − { ± 2 , ± 2 , ± 2 , …}
csc θ =
Dcsc
) א א
1
cos θ
= − { 0, ±π, ±2π, …}
sec θ =
Dsec
Rcsc = ( −∞, −1 ] ∪ [ 1, ∞ )
cos :
Rsec = ( −∞, −1 ] ∪ [ 1, ∞ )
8
8
4
4
4
4
8
8
٤
اﻟﺪوال اﻟﻤﺜﻠﺜﻴﺔ
اﻟﻌﻜﺴﻴﺔ
١{١{٢
sin : [ − π2 , π2 ] → [ −1,1 ] EאFאא sin−1 ½
sin−1 x = y ⇔
x = sin y
−1
sin : [ −1,1 ] → [ − π2 , π2 ]
cos : [ 0, π ] → [ −1,1 ] EאFאא cos−1 ½
cos−1 x = y ⇔
x = cos y
cos−1 : [ −1,1 ] → [ 0, π ]
tan : [
π π
−2,2
]→
EאFאא tan−1 ½
tan−1 x = y ⇔
x = tan y
−1
tan : → [ − π2 , π2 ]
2
1
3
2
1
0
1
1
1
1
0
1
2
2
1
10
5
0
5
10
1
2
sin ( x ) אא sin−1 ( x ) W
x ∈ [ −1,1 ] א cos−1 ( x ) sin−1 ( x ) א½ א
W x ∈ [ −1,1 ] ½
−1
sin ( sin x ) = x
cos ( cos−1 x ) = x
٥
θ WE sin θ = sin φ א φ ∈ [ − π2 , π2 ] אאאF sin−1 ( sin θ ) ½
− π2 ≤ θ ≤ 32π 2π א
sin
−1
⎧⎪ θ
( sin θ ) = ⎪
⎨
⎪⎪ π − θ
⎩
, − π2 ≤ θ ≤
, π2 ≤ θ ≤
π
2
3π
2
θ WE cos θ = cos φ א φ ∈ [ 0, π ] אאאF cos−1 ( cos θ ) ½
−π ≤ θ ≤ π 2π א
,0 ≤ θ ≤ π
⎧⎪ θ
⎪−
⎪⎩ θ
cos−1 ( cos θ ) = ⎪⎨
, −π ≤ θ ≤ 0
cos ( sin−1 −53 ) r cos−1 ( cos 7 ) q sin−1 ( sin 143π ) p sin ( sin−1 73 ) o W א٢
Kא
3
7
sin ( sin−1 73 ) = o
14 π
8π
− 2π =
3
3
>
8π
2π
− 2π =
3
3
sin−1 ( sin
14 π
3
3π
2
sin−1 ( sin 143π ) p
) = π − 23π
=
π
3
23π ∈ [ π2 , 32π ]
K 0 ≤ 7 ≤ π ، cos−1 ( cos 7 ) = 7
sin α =
q
א، α = sin−1 −53 W cos ( sin−1 −53 ) r
−3
5
, − π2 ≤ α ≤
π
2
Wאא α ، sin
٦
y
52 − 32 = 16 = 4
α
x
5
K cos α =
3
اﻟﻤﺠﺎور
4
=
اﻟﻮﺗــﺮ
5
، cos α
sin ( 2 cos−1 −41 ) p cos ( sin−1 −53 + tan−1 2 ) o Wא ٣
K o Kא
α = cos−1 −41 W sin ( 2 cos−1 −41 ) p
sin ( 2 cos−1 −41 ) = sin ( 2α ) = 2 sin α cos α
α = cos−1 −41 ⇔ cos α = −41 ,
0≤α≤π
אKאא α cos
42 − 12 = 15
y
4
α
x
1
sin ( 2 cos−1 −41 ) = 2
15
4
−1
15
=−
4
8
sin α =
( 415 )( )
W cos α =
−1
4
٧
cot−1 :
cot−1 x = y
sec−1 :
⇔
⇔
⇔
W sec−1 ½
x = sec y
\ ( −1,1 ) → ( −π, − π2 ] ∪ ( 0, π2 ]
csc−1 x = y
W cot−1 ½
x = cot y
\ ( −1,1 ) → [ 0, π2 ) ∪ [ π, 32π )
sec−1 x = y
csc−1 :
→ [ 0, π ]
W csc−1 ½
x = csc y
