Trigonometria no triângulo retângulo
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Trigonometria no triângulo retângulo
Razões trigonométricas
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Trigonometria no triângulo retângulo
Razões trigonométricas
F
E D C B A
G
H
I
ABI ~ ACH ~ ADG ~ AEF
AF
EF
AG
DG
AH
CH
AI
BI = cateto oposto
hipotenusasen =
AF
AE
AG
AD
AH
AC
AI
AB = cateto adjacente
hipotenusacos =
AE
EF
AD
DG
AC
CH
AB
BI = cateto oposto
cateto adjacentetg =
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Trigonometria no triângulo retângulo
Razões trigonométricas
Ângulos complementares
a
b
c
sen cos = c / a
sen cos = b / a
tg = tg1 = c / b
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Trigonometria no triângulo retângulotabela trigonométrica
d
a
a
45o
tg
cos
sen
30º 45º 60o
1
2
2
2
2
145
2
2
245cos45
a
atg
a
asen
o
oo
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Trigonometria no triângulo retângulotabela trigonométrica
tg
cos
sen
30º 45º 60o
1
2
2
2
2
h
60o
30oa a
a/2
sen60o = cos30o = 2
32
3
a
a
cos 60º = sen30o= 2
12 a
a
tg 60º = 3
2
2
3
a
a
tg30o = 3
3
3
1
60
1
otg
2
3
2
32
1
2
1
3
3 3
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Exercícios
1-Determine as medidas x e y dos lados dos triângulos abaixo.:
a)
5m
x
y
45o
b)
10m30o
x
y4m
xy
60o
c)
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Triângulo equilátero inscrito numa circunferência de raio R
R60o
l /2
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Triângulo equilátero circunscrito numa circunferência de raio R
R60o
l /2
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Comprimento da circunferência em radianos
R R R R R 0,28RR
C = 2 rad
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Equivalências
360º 2 rad
180º rad
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Transformações
a) 1º em rad.
b) 120º em rad
c) 6
5rad em graus
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A circunferência trigonométrica
x
y
1-1
1
-1
A
+
90o
180o
270o
360o
IQIIQ
IIIQ IVQ
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A circunferência trigonométrica
x
y
A
t
cos
sen tg
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A circunferência trigonométrica
x
y
A
t
cos
sen tg 1 sen2 + cos2 = 1
tg =sen
cos
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Simetria de arcos
A
180o -
180o + 360o -
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Simetria de arcos
1/2
2
3
30o150o
210o 330o
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Simetria de arcos
45o135o
225o315o
2
2
2
2
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Simetria de arcos
60o120o
240o 300o
1/2
2
3
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1
-1
A função seno
0
/2
3/2
2 /2
3/2 2 x
y
D = R
Im=[-1,1]
P = 2
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1
-1
A função cosseno
0
/2
3/2 2
/2
3/2 2 x
y
D = R
Im=[-1,1]
P = 2
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A função tangente
0
/2
3 /2
2
0/2
x
y