Question

Wykaż że jeżeli alfa (L), beta (B) i gamma (C) sa katami pewnego trojkata to sinus L +sinB + sinC= 4 cos (L/2 ) cos( B/2) cos (C/2)

Answer 1

Γ=180-(α+β)=2*[90-(α+β)/2]
sinα+sinβ+sinγ=2sin(α+β)/2*cos((α-β)/2+2sin[90-(α+β)/2]*cos[90-(α+β)/2]=
2sin(α+β)/2*cos((α-β)/2+2cos(α+β)/2* sin(α+β)/2=
2sin(α+β)/2*[cos(α-β)/2+cos(α+β)/2]=
4sin(α+β)/2*cosα/2*cosβ/2
ale
(α+β)/2=90-γ/2
wiec
sinα+sinβ+sinγ=4cosα/2 * cosβ/2 * cosγ/2

cbdu

wykorzystano wzory:
sin2α=2sinα*cosα
cosα+cosβ=2cos(α+β)/2 * cos(α-β)/2

pozdrawiam

Hnas