FORMULARIO DE TRIGONOMETRIA

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 𝑐𝑡𝑔 𝛼 =
𝐶.𝐴
𝐶.𝑂.
 𝑠𝑒𝑐 𝛼 =
𝐻
𝐶.𝐴.
 𝑐𝑠𝑐 𝛼 =
𝐻
𝐶.𝑂.
 𝑠𝑒𝑛 𝛼 =
𝐶.𝑂.
𝐻
 𝑐𝑜𝑠 𝛼 =
𝐶.𝐴.
𝐻
 𝑡𝑎𝑔 𝛼 =
𝐶.𝑂.
𝐶.𝐴
𝑎2 + 𝑏2 = 𝑐2 𝑃𝑖𝑡𝑎𝑔.
𝛼 + 𝛽 = 90° 𝐶𝑜𝑚𝑝.
𝑅𝑇 𝛼 = 𝐶𝑂_𝑅𝑇(𝛽)
0 ≤ 𝑠𝑒𝑛 𝛼 ≤ 1 𝑅𝑎𝑛𝑔.
𝑹𝑨𝒁𝑶𝑵𝑬𝑺 𝑻𝑹𝑰𝑮𝑶𝑵𝑶𝑴𝑬𝑻𝑹𝑰𝑪𝑨𝑺
𝑻𝑹𝑰𝑨𝑵𝑮𝑼𝑳𝑶𝑺 𝑵𝑶𝑻𝑨𝑩𝑳𝑬𝑺
30°
60° 2𝑘
𝑘
𝑘 3
37°
53° 5𝑘
3𝑘
4𝑘
45°
45° 2𝑘𝑘
𝑘
16°
74° 25𝑘
7𝑘
24𝑘
76°
14° 17𝑘
𝑘
4𝑘
8°
82° 5 2𝑘
𝑘
7𝑘
𝑨𝑵𝑮𝑼𝑳𝑶𝑺 𝑫𝑬 𝑬𝑳𝑬𝑽𝑨𝑪𝑰𝑶𝑵 𝒀 𝑫𝑬𝑷𝑹𝑬
𝛼
𝛽
Elevación
Depresión
Horizontal
13𝑘
5𝑘
12𝑘
15°
75° 4𝑘
( 6 + 2)𝑘
( 6 − 2)𝑘
𝑺𝑰𝑺𝑻𝑬𝑴𝑨 𝑫𝑬𝑴𝑬𝑫𝑰𝑪𝑰𝑶𝑵 𝑨𝑵𝑮𝑼𝑳𝑨𝑹
S
360°
=
C
400g
=
R
2π
ቐ
𝑆 → 𝑆𝑒𝑥𝑎𝑔𝑒𝑠𝑖𝑚𝑎𝑙
𝐶 → 𝐶𝑒𝑛𝑡𝑒𝑐𝑖𝑚𝑎𝑙
𝑅 → 𝑅𝑎𝑑𝑖𝑎𝑛
𝑇𝑎𝑚𝑏𝑖𝑒𝑛
S
9°
=
C
10g
=
20R
π
𝑆 ቊ1° = 60
′
1′ = 60′′
𝐶 ቊ
1𝑔 = 100𝑚
1𝑚 = 100𝑠
1 𝑣𝑢𝑒𝑙𝑡𝑎 ቐ
360°
400𝑔
2𝜋
𝑺𝑰𝑺𝑻𝑬𝑴𝑨 𝑪𝑶𝑶𝑹𝑫𝑬𝑵𝑨𝑫𝑨𝑺 𝑪𝑨𝑹𝑻𝑬𝑺.
𝑑
𝑎
𝑏
(𝑥1, 𝑦1)
(𝑥2, 𝑦2)
(𝑥3, 𝑦3)
𝑑 = (𝑥1 − 𝑥2)
2+(𝑦1 − 𝑦2)
2
𝑥3 =
𝑥1 𝑎 + 𝑥2(𝑏)
𝑎 + 𝑏
𝑦3 =
𝑦1 𝑎 + 𝑦2(𝑏)
𝑎 + 𝑏
𝑆𝑖: 𝑎 = 𝑏
𝑥3 =
𝑥1 + 𝑥2
2
𝑦3 =
𝑦1 + 𝑦2
2
𝑺𝑰𝑺𝑻𝑬𝑴𝑨 𝑪𝑶𝑶𝑹𝑫𝑬𝑵𝑨𝑫𝑨𝑺 𝑪𝑨𝑹𝑻𝑬𝑺.
(𝑥1, 𝑦1)
𝑆 =
𝐴 − 𝐵
2
(𝑥2, 𝑦2)
(𝑥3, 𝑦3) 𝑥1 𝑦1
𝑥2 𝑦2
𝑥3 𝑦3
𝑥1 𝑦1
𝑥1𝑦2
𝑥2𝑦3
𝑥3𝑦1
𝑥2𝑦1
𝑥3𝑦2
𝑥1𝑦3
+ +
𝐵 𝐴
𝜃
𝛼
𝛽
𝐿𝑎𝑑𝑜 𝑓𝑖𝑛𝑎𝑙
𝐿𝑎𝑑𝑜 𝑓𝑖𝑛𝑎𝑙
𝐿𝑎𝑑𝑜 𝑖𝑛𝑖𝑐𝑖𝑎𝑙
𝜃 𝑦 𝛼 𝑒𝑛 𝑃. 𝑁.
𝜃 𝑦 𝛽 𝐶𝑜𝑡𝑒𝑟𝑚𝑖𝑛
𝜃 > 0 𝛽 < 0 𝛼 < 0
𝑹𝑬𝑫𝑼𝑪𝑪𝑰𝑶𝑵 𝑨𝑳 𝑷𝑹𝑰𝑴𝑬𝑹 𝑪𝑼𝑨𝑫𝑹𝑨𝑵
𝑅𝑇
180° ± 𝛼
360° ± 𝛼
= ±𝑅𝑇(𝛼) 𝑅𝑇
90° ± 𝛼
270° ± 𝛼
= ±𝐶𝑂𝑅𝑇(𝛼)
𝑅𝑇 360°𝑛 + 𝛼 = 𝑅𝑇 𝛼  cos −𝛼 = cos𝛼
 sen −𝛼 = −sen 𝛼
 tan −𝛼 = −tan𝛼
𝑺𝒊: 𝛼 + 𝛽 = 180°
𝑠𝑒𝑛 𝛼 = 𝑠𝑒𝑛𝛽
𝑐𝑜𝑠 𝛼 = −𝑐𝑜𝑠𝛽
𝑡𝑎𝑛 𝛼 = − tan𝛽
𝑛 → #𝑣𝑢𝑒𝑙𝑡𝑎𝑠
𝑺𝒊: 𝛼 + 𝛽 = 360°
𝑠𝑒𝑛 𝛼 = −𝑠𝑒𝑛𝛽
𝑐𝑜𝑠 𝛼 = 𝑐𝑜𝑠𝛽
𝑡𝑎𝑛 𝛼 = − tan𝛽
𝐴𝑛𝑔𝑢𝑙𝑜𝑠
𝑛𝑒𝑔𝑎𝑡𝑖𝑣
𝑃𝑟𝑜𝑝.
𝑰𝑫𝑬𝑵𝑻𝑰𝑫𝑨𝑫𝑬𝑺 𝑨𝑹𝑪𝑶 𝑺𝑰𝑴𝑷𝑳𝑬.
 𝑠𝑒𝑛 𝑥 csc 𝑥 = 1
 𝑐𝑜𝑠 𝑥 sec 𝑥 = 1
 𝑡𝑎𝑛 𝑥 cot 𝑥 = 1
 tan 𝑥 =
𝑠𝑒𝑛 𝑥
cos 𝑥
∗ cot 𝑥 =
𝑐𝑜𝑠 𝑥
𝑠𝑒𝑛 𝑥
 𝑠𝑒𝑛2 𝑥 + cos2 𝑥 = 1
 sec2 𝑥 − tan2 𝑥 = 1
 csc2 𝑥 − cot2 𝑥 = 1
 𝑠𝑒𝑛4 𝑥 + cos4 𝑥 = 1 − 2𝑠𝑒𝑛2𝑥 𝑐𝑜𝑠2𝑥
 𝑠𝑒𝑛6 𝑥 + cos6 𝑥 = 1 − 3𝑠𝑒𝑛2𝑥 𝑐𝑜𝑠2𝑥
 tan 𝑥 + cot 𝑥 = sec 𝑥 csc 𝑥
 sec2 𝑥 + csc2 𝑥 = sec2 𝑥 csc2 𝑥
 (1 ± 𝑠𝑒𝑛 𝑥 ± cos 𝑥)2= 2(1 ± 𝑠𝑒𝑛 𝑥)(1 ± cos 𝑥)
𝑰𝑫𝑬𝑵𝑻𝑰𝑫𝑨𝑫𝑬𝑺 𝑨𝑹𝑪𝑶 𝑪𝑶𝑴𝑷𝑼𝑬𝑺𝑻𝑶
 sen α ± β = sen α cos𝛽 ± cos𝛼 𝑠𝑒𝑛 𝛽
 cos α ± β = cos α cos𝛽 ∓ sen 𝛼 𝑠𝑒𝑛 𝛽
 tan α ± β =
tan 𝛼±tan 𝛽
1∓tan 𝛼 tan 𝛽
 cot α ± β =
cot 𝛼 cot 𝛽±1
cot 𝛼±cot 𝛽
 sen α + β sen α − β = sen2α − sen2β
 cos α + β cos α − β = cos2α − sen2β
 tan 𝛼 ± tan 𝛽 =
𝑠𝑒𝑛 (𝛼±𝛽)
cos 𝛼 cos 𝛽
𝑺𝒊 𝐴 + 𝐵 + 𝐶 = 𝜋
tan𝐴 + tan𝐵 + tan𝐶 = tan𝐴 tan𝐵 tan𝐶
cot𝐴 cot𝐵 + cot𝐴 cot 𝐶 + cot𝐵 cot 𝐶 = 1
𝑰𝑫𝑬𝑵𝑻𝑰𝑫𝑨𝑫𝑬𝑺 𝑨𝑹𝑪𝑶𝑴𝑬𝑫𝑰𝑶
𝑰𝑫𝑬𝑵𝑻𝑰𝑫𝑨𝑫𝑬𝑺 𝑨𝑹𝑪𝑶 𝑫𝑶𝑩𝑳𝑬.
𝑰𝑫𝑬𝑵𝑻𝑰𝑫𝑨𝑫𝑬𝑺 𝑨𝑹𝑪𝑶 𝑻𝑹𝑰𝑷𝑳𝑬
 sen
α
2
= ±
1−𝐶𝑂𝑆 𝛼
2
 cos
α
2
= ±
1+𝐶𝑂𝑆 𝛼
2
 sen
𝛼
2
+ cos
𝛼
2
= ± 1 + 𝑠𝑒𝑛 𝛼
 sen
𝛼
2
− cos
𝛼
2
= ± 1 − 𝑠𝑒𝑛 𝛼
 tan
α
2
= ±
1−𝐶𝑂𝑆 𝛼
1+𝐶𝑂𝑆 𝛼
 tan
𝛼
2
= csc𝛼 − cot𝛼
 cot
𝛼
2
= csc 𝛼 + cot 𝛼
 sen 2α = 2sen α cosα
 cos 2α = cos2 α − sen2 𝛼
 tan 2α =
2tan 𝛼
1−tan2 α
 2 sen α = 1 − cos 2𝛼
 2 cosα = 1 + cos2𝛼
 cot α + tanα = 2 csc2α
 cot α + tanα = 2 cot 2α
 𝑠𝑒𝑛4 α + cos4 α =
3
4
+
1
4
cos 4α
 𝑠𝑒𝑛6 α + cos6 α =
5
8
+
3
8
cos 4α
2𝛼
2
ta
n
𝛼
1 − 𝑡𝑎𝑛2 𝛼
sen 3𝑥 = 3 sen x − 4 sen3 x
cos 3𝑥 = 4 cos3 𝑥 − 3 cos 𝑥
tan 3𝑥 =
3 tan 𝑥−tan3 𝑥
1−3 tan2 𝑥
 4 sen3 𝑥 = 3 𝑠𝑒𝑛 𝑥 − 𝑠𝑒𝑛 3𝑥
 4 cos3 𝑥 = 3 𝑐𝑜𝑠 𝑥 + cos 3𝑥
 sen 3x = sen x (2 cos(2𝑥) + 1)
 cos 3x = cos x (2 cos 2𝑥 − 1)
 sen 3x = 4sen x sen 60° − x sen(60° + x)
 cos 3x = 4cos x cos 60° − x cos(60° + x)
 tan 3𝑥 = tan 𝑥 tan 60° − x tan(60° + x)
𝑭𝑼𝑵𝑪𝑰𝑶𝑵 𝑺𝑬𝑵𝑶
𝑭𝑼𝑵𝑪𝑰𝑶𝑵 𝑪𝑶𝑺𝑬𝑵𝑶.
𝑭𝑼𝑵𝑪𝑰𝑶𝑵 𝑻𝑨𝑵𝑮𝑬𝑵𝑻𝑬
𝑭𝑼𝑵𝑪𝑰𝑶𝑵 𝑪𝑶𝑻𝑨𝑵𝑮𝑬𝑵𝑻𝑬
𝑭𝑼𝑵𝑪𝑰𝑶𝑵 𝑺𝑬𝑪𝑨𝑵𝑻𝑬
𝑫𝑶𝑴𝑰𝑵𝑰𝑶 𝒀 𝑹𝑨𝑵𝑮𝑶
𝑻𝑹𝑨𝑺𝑭𝑶𝑹𝑴𝑨𝑪𝑰𝑶𝑵𝑬𝑺 𝑻𝑹𝑰𝑮𝑶𝑵𝑶𝑴𝑬
 sen𝐴 + 𝑠𝑒𝑛 𝐵 = 2sen
𝐴+𝐵
2
cos
𝐴−𝐵
2
 sen𝐴 − 𝑠𝑒𝑛 𝐵 = 2cos
𝐴+𝐵
2
sen
𝐴−𝐵
2
 cos𝐴 + 𝑐𝑜𝑠 𝐵 = 2cos
𝐴+𝐵
2
cos
𝐴−𝐵
2
 cos𝐴 − 𝑐𝑜𝑠 𝐵 = −2sen
𝐴+𝐵
2
sen
𝐴−𝐵
2
 2sen𝐴𝑐𝑜𝑠𝐵 = 𝑠𝑒𝑛 𝐴 + 𝐵 + 𝑠𝑒𝑛(𝐴 − 𝐵)
 2cos𝐴𝑠𝑒𝑛𝐵 = 𝑠𝑒𝑛 𝐴 + 𝐵 − 𝑠𝑒𝑛(𝐴 − 𝐵)
 2cos𝐴𝑐𝑜𝑠𝐵 = 𝑐𝑜𝑠 𝐴 + 𝐵 + 𝑐𝑜𝑠(𝐴 − 𝐵)
 2sen𝐴𝑠𝑒𝑛𝐵 = 𝑐𝑜𝑠 𝐴 − 𝐵 − 𝑠𝑒𝑛(𝐴 + 𝐵)
Ide. Auxil: Si: 𝐴 + 𝐵 + 𝐶 = 180°, se cumple:
 𝑠𝑒𝑛𝐴 + 𝑠𝑒𝑛𝐵 + 𝑠𝑒𝑛𝐶 = 4 cos
𝐴
2
cos
𝐵
2
cos
𝐶
2
 𝑐𝑜𝑠𝐴 + 𝑐𝑜𝑠𝐵 + 𝑐𝑜𝑠𝐶 = 4sen
𝐴
2
sen
𝐵
2
sen
𝐶
2
+ 1
 𝑠𝑒𝑛2𝐴 + 𝑠𝑒𝑛2𝐵 + 𝑠𝑒𝑛2𝐶 = 4𝑠𝑒𝑛𝐴𝑠𝑒𝑛𝐵𝑠𝑒𝑛𝐶
 𝑐𝑜𝑠2𝐴 + 𝑐𝑜𝑠2𝐵 + 𝑐𝑜𝑠2𝐶 = −4𝑐𝑜𝑠𝐴𝑐𝑜𝑠𝐵𝑐𝑜𝑠𝐶 − 1
𝑇𝑟𝑎𝑛𝑠. 𝑆𝑢𝑚𝑎
𝑜 𝑟𝑒𝑠𝑡𝑎
𝑎 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑜
𝑇𝑟𝑎𝑛𝑠. 𝑑𝑒
𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑜
𝑎 𝑠𝑢𝑚𝑎 𝑜
𝑟𝑒𝑠𝑡𝑎
𝑳𝑰𝑵𝑬𝑨𝑺 𝑻𝑹𝑰𝑮𝑶𝑵𝑶𝑴𝑬𝑻𝑹𝑰𝑪𝑨𝑺
𝑉
𝑆
𝑇
𝑈𝐴𝑄𝑂
𝑅
𝐵
𝑃
𝑌
𝑋
𝛼
𝑠𝑒𝑛 𝛼 = 𝑃𝑄
𝑐𝑜𝑠 𝛼 = 𝑅𝑃
𝑡𝑎𝑛 𝛼 = 𝑆𝐴
𝑐𝑜𝑡 𝛼 = 𝐵𝑇
𝑠𝑒𝑐 𝛼 = 𝑂𝑈
𝑐𝑠𝑐 𝛼 = 𝑂𝑉
𝑭𝑼𝑵𝑪𝑰𝑶𝑵𝑬𝑺 𝑻𝑹𝑰𝑮𝑶𝑵𝑶𝑴.𝑹𝑬𝑨𝑳𝑬𝑺
𝑦
𝑥
𝐴
𝑚
𝑝
𝑙𝑖
𝑡𝑢
𝑑
𝐴
𝑚
𝑝
𝑙𝑖
𝑡𝑢
𝑑
𝑃𝑒𝑟𝑖𝑜𝑑𝑜
𝑦 = 𝑓 𝑥 = 𝑎 𝑠𝑒𝑛 (𝑏𝑥 + 𝑐)
𝑇𝑟𝑎𝑠𝑙𝑎𝑐𝑖ó𝑛
𝑎
𝑎
2𝜋/𝑏
𝑐
FUNCION
Y= rt(X)
DOMINIO 𝑥 ∈ RANGO
𝑦 = 𝑠𝑒𝑛 𝑥 𝑅 𝑦 ∈ [−1; 1]
𝑦 = 𝑐𝑜𝑠 𝑥 𝑅 𝑦 ∈ [−1; 1]
𝑦 = 𝑡𝑎𝑛 𝑥 𝑅 − {(2𝑛 + 1)
𝜋
2
} 𝑦 ∈ 𝑅
𝑦 = 𝑐𝑜𝑡 𝑥 𝑅 − {(𝑛)𝜋} 𝑦 ∈ 𝑅
𝑦 = 𝑠𝑒𝑐 𝑥 𝑅 − {(2𝑛 + 1)
𝜋
2
} 𝑦 ∈ 𝑅− < −1; 1 >
𝑦 = 𝑐𝑠𝑐 𝑥 𝑅 − {(𝑛)𝜋} 𝑦 ∈ 𝑅− < −1; 1 >