В треугольнике со сторонами: a = 5, b = 4, с = 2.522, углы равны α° = 97.52°, β° = 52.47°, γ° = 30°
Ответ:
a=5
b=4
c=2.522
α°=97.52°
β°=52.47°
γ°=30°
S = 5.001
ha=2
hb=2.501
hc=3.966
P = 11.52
Решение:
Сторона:
c = √a2 + b2 - 2ab·cos(γ°)
= √52 + 42 - 2·5·4·cos(30°)
= √25 + 16 - 40·0.866
= √6.36
= 2.522
Угол:
α° = arcsin(
a
c
sin(γ°))
= arcsin(
5
2.522
sin(30°))
= arcsin(1.983·0.5)
= 82.52°
или:
α° = arccos(
b2+c2-a2
2bc
)
= arccos(
42+2.5222-52
2·4·2.522
)
= arccos(
16+6.360484-25
20.18
)
= 97.52°
Угол:
β° = arcsin(
b
c
sin(γ°))
= arcsin(
4
2.522
sin(30°))
= arcsin(1.586·0.5)
= 52.47°
Периметр:
P = a + b + c
= 5 + 4 + 2.522
= 11.52
Площадь:
p =
a + b + c
2
S =√p·(p-a)·(p-b)·(p-c)
=√5.761·(5.761-5)·(5.761-4)·(5.761-2.522)
=√5.761 · 0.761 · 1.761 · 3.239
=√25.006495705359
= 5.001
Высота :
ha =
2S
a
=
2 · 5.001
5
= 2
hb =
2S
b
=
2 · 5.001
4
= 2.501
hc =
2S
c
=
2 · 5.001
2.522
= 3.966