θ
cos θ = Abcisa deP sen θ = Ordenada deP
|S |
AP AP
cos S = = Abcisa deP sen S = = Ordenada deP
f (x) = g
2πx 360
,
g (x) = f
360x 2π
90◦ f (90) =
2π(90) π 180 π = = 360 360 2
π
2
360 π2 360(π ) = = = 90 g 2 2π 4π π
θ = θ + 360
cos(x + 360) = cos( x) y sen(x + 360) = sen(x),
cos θ
∀ x
sen θ
cos θ = Abcisa de P
sen θ = Ordenada de P
−1 ≤ Abcisa de P ≤ ≤ 1 |cos θ| ≤ 1 −1 ≤ Ordenada de P ≤ ≤ 1 |sen θ| ≤ 1
cos(x + 2π ) = cos(x)
y sen(x + 2 π) = sen( x),
cos θ = Abcisa de P cos θ = 0
⇔
θ =
±π
∀ x
sen θ = Ordenada de P sen θ = 0
tan θ =
2
⇔
θ =
± π2
sen θ 1 1 cos θ , sec θ = , csc θ = , cot θ = cos θ cos θ sen θ sen θ
2
b + a = c
2
⇒
b2 a 2 + 2 =1 c2 c
⇒ b c
2
+
a c
2
=1
cos θ =
b a y sen θ = c c
⇒
c cos θ = b y c sen θ = a
45 a = b =
√ 12 y c = 1 ⇒ cos 45 = √ 12 ,
sen 45 =
√ 12 ,
tan 45 = 1, cot 45 = 1, sec 45 =
P = (x, y )
√
2, csc 45 =
θ
cos θ = x, y sen θ = y π
P = (y, x)
2
cos
−
⇒
cos
−
sen
−
⇒
sen
−
π
2
π
2
θ = y
θ = x
π
2
π
2
θ = sen θ
θ = cos θ π
P = ( y, x)
−
cos
2
− π
+ θ =
2
sen
π
2
⇒
cos
+ θ = x
⇒
sen
−
cos(π
−
y
P = ( x, y )
− θ
π
2
π
2
θ =
+ θ
− sen θ
+ θ = cos θ π
−θ
− θ) = −x ⇒ cos(π − θ) = − cos θ
sen(π
− θ) = y ⇒ sen(π − θ) = sen θ
√
2
P = ( x,
− −y)
π + θ
cos(π + θ ) =
−x ⇒ cos(π + θ) = − cos θ
sen(π + θ ) =
−y ⇒ sen(π + θ) = − sen θ
P = (x, y )
π + θ
−
cos( θ) = x
−
sen( θ) =
−
⇒ cos(−θ) = cos θ
−y ⇒ sen(−θ) = − sen θ
cos θ sen θ
− − sen θ = − tan θ − cos θ cos(−θ) cos θ cot(−θ) = =− = − cot θ sen(−θ) sen θ sen − θ cos θ π tan − θ = = = cot θ 2 sen θ cos − θ tan( θ) =
−
− π
cot
2
sen( θ) = cos( θ)
π 2
π 2
θ =
cos sen
sen tan + θ = 2 cos π
cot
π
2
+ θ =
− − −−
cos sen
π 2
π 2
π 2
π 2
π 2
π 2
θ
θ
=
sen θ = tan θ cos θ
+ θ cos θ = = sen θ + θ
− cot θ
+ θ sen θ = = cos θ + θ
− tan θ
sen θ π − θ) − θ) = sen( = − cos θ = − tan θ cos(π − θ) cos(π − θ) − cos θ cot(π − θ) = = = − cot θ sen(π − θ) sen θ
tan(π
2
= cos2
2
2
|P Q| = (cos A − cos B) + (sen A − sen B) A − 2cos A cos B + sen −2sen A sen B + cos B + sen = 2 − 2(cos B cos A + sen A sen B ) 2
2
|P Q| 2
=2
2
2
B
− 2 cos( cos(A − B )
− 2cos(A − B) = 2 − 2(cos B cos A + sen A sen B) cos(A
− B) = cos B cos A + sen A sen B −B cos(A + B ) = cos(A − (−B )) = cos(−B )cos A + sen A sen(−B ) = cos B cos A − sen A sen B B por B +
π
2
sen(A + B ) = sen A cos B + cos A sen B
(θ, sen θ ) P θ
0
θ
2π
P θ π
3π 2
2π
3π 2
P θ
cos θ = sen
θ +
π
2
0
1
π
1
π
θ
sen
cos tan θ =
Domtan θ = R
−
sen θ cos θ
∈
(2k + 1) π k 2
Z
θ
1
cos sec θ =
Domtan θ = R
−
1 cos θ
∈
(2k + 1) π k 2
Z
θ
1
tan cot θ =
Domcot θ =
R
−
cos θ sen θ
∈ (2k ) pi k
Z
θ
1
sen sec θ =
Domcsc θ = R
−
1 sen θ
∈ (2k ) pi k
Z
− → − → −◦ π π , 2 2 π π ,
f :
f −1 [ 1, 1]
−
[ 1, 1]
f (x) = sen(x)
2 2 sen−1 (sin(x)) = sen−1 sin (x) = (x) = x
sen−1 (y ) = x
sen(sen−1 (x) = (sen sen−1 )(x) = (x) = x
◦
sin−1 ( √
−√ 1 ≤
⇒ sin(y) =
3 2
3 2
√
3
x
)
≤1
sin
π 3
=
√
y = sin−1 (
3 2
√
3 2
1
)
⇒ sin(y) = sin(sin− ( ⇒ y = π3 sin(y) = sin π 3
f : [0, π ] [ 1, 1] f (x) = cos(x) − 1 − 1 [0, π] cos (y) = x f [ 1, 1] cos−1 (cos(x)) = cos−1 cos (x) = (x) = x
−
→ −
→
◦
cos(cos−1 (x) = (cos cos−1 )(x) = (x) = x
◦
√
3 2
))
⇒
tan(x) = y
⇔
tan−1 (y ) = x
Domarctan = (
cot(x) = y
⇔
−∞, ∞) imagen =
− π π ,
2 2
cot−1 (y ) = x
Domcot− = [
−∞, ∞] imagen = (0, π)
1
csc(x) = y Domcsc−
1
1
⇔ csc− (y) = x π = [ −∞, −1] ∪ [1, ∞] imagen = − , 0 ∪, 2
0,
π
2
sec(x) = y
⇔
Domsec− = [ 1
sec−1 (y) = x
−∞, −1] ∪ [1, ∞] imagen =
∪ 0,
π
2
,
π
2
,π
