Trigonométrie et calcul numérique

sin2A+ sin2B+ sin2C= 2
√3 cos 2x+ sin 2x= 2
α β γ
θ
4m2−a2= 4 b c cos α S =a m
2sin θ
c= 3,45 α= 48◦
β= 73◦θ

sin2A+ sin2B+ sin2C= 2
sin2A+ sin2B+ sin2C= 2
A+B+C=π C =π−(A+B)
sin C= sin(π−(A+B)) = sin(A+B)
sin2A+ sin2B+ sin2C= 2
⇔sin2A+ sin2B+ sin2(A+B) = 2
⇔sin2A+ sin2B+ (sin Acos B+ cos Asin B)2= 2
⇔sin2A+ sin2B+ sin2Acos2B+ cos2Asin2B+ 2 sin Acos Asin Bcos B= 2
sin2A= 1 −cos2A
sin2B= 1 −cos2B
⇔1−cos2A+ 1 −cos2B+ cos2B−cos2Acos2B+ cos2A−cos2Acos2B
+ 2 sin Acos Asin Bcos B= 2
⇔ − cos2Acos2B−cos2Acos2B+ 2 sin Acos Asin Bcos B= 2
⇔2 cos Acos B(sin Asin B−cos Acos B)=0
⇔2 cos Acos Bcos(A+B)=0

cos(A+B) = cos(π−C) = −cos C
⇔2 cos Acos Bcos C= 0
]0, π[
A=π
2B=π 2C=π 2
sin2A+ sin2B+ sin2C= 2
A=π
2
sin A= 1
A+B+C=π
B=π
2−Csin B= sin(π 2−C) = cos C
sin2A+ sin2B+ sin2C= 2
⇔1 + cos2C+ sin2C= 2
cos2C+ sin2C= 1
sin2A+ sin2B+ sin2C= 2 ⇔cos Acos Bcos C= 0
A=π
2cos A= 0
√3 cos 2x+ sin 2x= 2
√3 cos 2x+ sin 2x= 2
⇔
√3
2cos 2x+1 2sin 2x= 1

√3
2= cos π 6
1
2= sin π 6
⇔cos π
6cos 2x+ sin π 6sin 2x= 1
⇔cos(2x−π
6)=1
2x−π
6= 0 + 2 k π
⇔2x=π
6+ 2 k π
⇔x=π
12 +k π
x= 15◦+k180◦
α β γ
θ
4m2−a2= 4 b c cos α S =a m
2sin θ
c= 3,45 α= 48◦
β= 73◦θ

A
B C
M
q
m
cb
a
a
bg
AB2=AM 2+MB2−2 cos θ AM MB
⇔c2=a2
4+m2−2a 2mcos θ
AC2=AM2+MC2−2 cos(π−θ)AM MC
⇔b2=m2+a2
4+ 2 a 2mcos θ
b2+c2= 2 m2+a2
2
a2=b2+c2−2b c cos α
b2+c2
a2=b2+c2−2b c cos α
⇔a2= 2 m2+a2
2−2b c cos α
4m2−a2= 4 b c cos α
S=a h 1
2