Sinus et cosinus

Question 1

mesure principale

Answer 1

On appelle mesure principale d’un angle α, la mesure x qui se trouve dans l’intervalle ]−π;π]

Question 2

Equations trigonométriques

Answer 2

•L’équation cosx = cosa admet les solutions suivantes sur R:

x=a+k2π ou x=−a+k2π avec k∈Z

•L’équation sinx=sina admet les solutions suivantes sur R:

x=a+k2π ou x=π−a+k2π avec k∈Z

Question 3

sinx > 0

Answer 3

x∈ ]0 ;π[

Question 4

cosx > 0

Answer 4

x∈ ]−π2;π2[

Question 5

définition

Answer 5

À tout réel x, on associe un point unique M du cercle unité ou cercle trigonométrique de centre O, dont les coordonnées sont :

M(cosx; sinx)

Question 6

parité

Answer 6

• La fonction sinus est impaire : ∀x∈R sin(−x) =−sinx

* La fonction cosinus est paire : ∀x∈R cos(−x) =cosx

Question 7

périodicité

Answer 7

f est periodique de periode T si sur un itervalle I:
x+ T appartient à I
f(x+T)=f(x)

Question 8

sin(π/2−x)

Answer 8

sin(π/2−x)=cosx

Question 9

cos(π/2−x)

Answer 9

cos(π/2−x)=sinx

Question 10

sin′x

Answer 10

cosx

Question 11

cos’x

Answer 11

-sinx

Question 12

x=0

Answer 12

cos0=1 sin0=0

Question 13

x=π/6

Answer 13

cosπ/6=racine de 3/2

sinπ/6= 1/2

Question 14

x=π/4

Answer 14

cos et sin: racine de 2/2

Question 15

x=π/3

Answer 15

cosπ/3=1/2

sinπ/3= racine de 3/2

Question 16

x=π/2

Answer 16

cos1/2=0

sin1/2=1

Question 17

x=π

Answer 17

cosπ=-1

sinπ=0

Question 18

somme des carrés d’un cos et d’un sin

Answer 18

cosx²+sinx²=1

Question 19

sin(π/2+x)

Answer 19

cosx

Question 20

cos(π/2+x)

Answer 20

-sinx

Question 21

cos(a-b)

Answer 21

cosa cosb + sina sinb

Question 22

cos(a+b)

Answer 22

cosa cosb - sina sinb

Question 23

sin (a-b)

Answer 23

cosa sinb - cosb sina

Question 24

sin (a+b)

Answer 24

cosa sinb + cosb sina

Question 25

cos2x

Answer 25

= cos²x-sin²x = 2cos²x-1 = 1-2sin²x

Question 26

sin2x

Answer 26

2sinxcosx

Question 27

1 + tan²(x)

Answer 27

1 + tan²(x) = 1/cos2(x)

Question 28

tan(a + b)

Answer 28

tan(a + b) = tan(a) + tan(b)/1 − tan(a) tan(b)

Question 29

tan(a − b)

Answer 29

tan(a − b) = tan(a) − tan(b)/1 + tan(a) tan(b)

Question 30

tan(2a)

Answer 30

tan(2a) = 2 tan(a)/1 − tan²(a)

Question 31

cos²a

Answer 31

cos²(a) = 1 + cos(2a)/2

Question 32

sin²(a)

Answer 32

sin²(a) = 1 − cos(2a)2

Question 33

cos(a) cos(b)

Answer 33

cos(a) cos(b) = 1/2(cos(a − b) + cos(a + b))

Question 34

sin(a) sin(b)

Answer 34

sin(a) sin(b) = 1/2(cos(a − b) − cos(a + b))

Question 35

sin(a) cos(b)

Answer 35

sin(a) cos(b) = 1/2(sin(a − b) + sin(a + b))

Question 36

on pose t = tan (x/2)
tan(x)
sin(x)
cos(x)

Answer 36

tan(x) = 2t/1 − t²

sin(x) = 2t/1 + t²

cos(x) = 1 − t²/1 + t²