Răspuns:
Explicație pas cu pas:
a)
cos²(a) + sin²(a) = 1
cos²(a) = 1 - sin²(a)
cos(a) = √[1 - sin²(a)] = √[1- (3/5)²] = √(1 - 9/25) = √(25/25 - 9/25) = √[(25-9)/25] = √(16/25) = 4/5
b)
sin(a+b) = sin(a)*cos(b) + cos(a)*sin(b)
cos²(b) + sin²(b) = 1
cos²(b) = 1 - sin²(b)
cos(b) = √[1 - sin²(b)] = √[1- (5/13)²] = √(1 - 25/169) = √(169/169 - 25/169) = √[(169 -25)/169] = √(144/169) = 12/13
sin(a+b) = sin(a)*cos(b) + cos(a)*sin(b = (3/5)*(12/13)*(4/5)*(5/13) = (3*4*5*12)/(5*5*13*13) = 144/845
c)
Notam sin(x) = y
[tex][\sqrt{1+ sin(x)} +\sqrt{1 -sin(x)}]^2+[\sqrt{1+ sin(x)} -\sqrt{1 -sin(x)}]^2 =[/tex]
[tex](\sqrt{1+ y} +\sqrt{1 -y})^2+(\sqrt{1+y} -\sqrt{1-y})^2 =[/tex]
[tex]=(\sqrt{1+ y})^2 +2*\sqrt{1+y}*\sqrt{1- y} +(\sqrt{1- y})^2+ (\sqrt{1+ y})^2 -2*\sqrt{1+y}*\sqrt{1- y} +(\sqrt{1- y})^2 =[/tex]
[tex]=1+ y +2*\sqrt{1+y}*\sqrt{1- y} +1- y+ 1+ y-2*\sqrt{1+y}*\sqrt{1- y} +1- y = 4[/tex]
