RÄspuns :
2sin(a/2)cos(a/2)=3/5
notÄm x =cos(a/2)
2*ā(1-x^2)* x= 3/5
(1-x^2)*x^2=9/100
y=x^2
y-y^2-9/100=0
100yĀ°2-100y+9=0
y1=[100-ā(10000-4*100*9)]/200=1/10
y-2=180/200=9/10
x1=1/ā10. x2=-1/ā10
X3=3/ā10. x-4= -3/ā10
cum a/2<pi/2
sunt valabile doar valorile pozitive ( deci 2 valori!)
notÄm x =cos(a/2)
2*ā(1-x^2)* x= 3/5
(1-x^2)*x^2=9/100
y=x^2
y-y^2-9/100=0
100yĀ°2-100y+9=0
y1=[100-ā(10000-4*100*9)]/200=1/10
y-2=180/200=9/10
x1=1/ā10. x2=-1/ā10
X3=3/ā10. x-4= -3/ā10
cum a/2<pi/2
sunt valabile doar valorile pozitive ( deci 2 valori!)
[tex]\it a\in\Big(\dfrac{\pi}{2},\ \pi\Big) \Rightarrow \cos a < 0\\ \\ \\ \cos a =-\sqrt{1-sin^2a} =-\sqrt{1-\dfrac{9}{25}}=-\sqrt{\dfrac{16}{25}}=-\dfrac{4}{5}\\ \\ \\ a\in\Big(\dfrac{\pi}{2},\ \pi\Big) \Rightarrow \dfrac{a}{2} \in\Big(0,\ \dfrac{\pi}{2}\Big) \Rightarrow cos\dfrac{a}{2}>0\ \ \ \ \ (*)\\ \\ \\ -\dfrac{4}{5} =\cos a = \cos 2\cdot\dfrac{a}{2} =2cos^2\dfrac{a}{2}-1 \Rightarrow cos^2\dfrac{a}{2}=\dfrac{\dfrac{1}{5}}{2} \Rightarrow cos^2\dfrac{a}{2} =\dfrac{1}{10}\stackrel{(*)}{\Longrightarrow}[/tex]
[tex]\it \Rightarrow cos\dfrac{a}{2} =\dfrac{1}{\sqrt{10}}=\dfrac{\sqrt{10}}{10}[/tex]
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