Given that α and β are two distinct acute angles satisfying the equation 4cosθ^2-2cosθ-1=0.
(a) find the values of cosα+cosβ and cosαcosβ.
(b) using the fact that cos(x/2)^2=(1+cosx)/2,or otherwise,prove that cos((α+β)/2)+cos((α-β)/2)=√5/2
a.maths
2007-07-19 4:45 am
回答 (2)
2007-07-19 5:05 am
參考: Myself~~~
2007-07-19 5:13 am
a)
since
4cos²A - 2cosA - 1 = 0 and 4cos²B - 2cosB - 1 = 0
cosA and cosB is the root of 4x² - 2x - 1 = 0
so,
cosA + cosB = -(-2)/4 = 1/2
cosAcosB = -1/4
b)
cos((A + B)/2) + cos((A - B)/2)
= sqrt((cos²((A + B)/2) + cos²((A - B)/2) + 2[cos((A + B)/2)][cos((A - B)/2))]
= sqrt((1/2)[1 + cos(A + B) + 1 + cos(A - B)] + 2(1/2)[cosA + cosB])
= sqrt((1/2)[2 + 2cosAcosB] + cosA + cosB]
= sqrt([1 + (-1/4)] + (1/2))
= sqrt(5)/2
since
4cos²A - 2cosA - 1 = 0 and 4cos²B - 2cosB - 1 = 0
cosA and cosB is the root of 4x² - 2x - 1 = 0
so,
cosA + cosB = -(-2)/4 = 1/2
cosAcosB = -1/4
b)
cos((A + B)/2) + cos((A - B)/2)
= sqrt((cos²((A + B)/2) + cos²((A - B)/2) + 2[cos((A + B)/2)][cos((A - B)/2))]
= sqrt((1/2)[1 + cos(A + B) + 1 + cos(A - B)] + 2(1/2)[cosA + cosB])
= sqrt((1/2)[2 + 2cosAcosB] + cosA + cosB]
= sqrt([1 + (-1/4)] + (1/2))
= sqrt(5)/2
收錄日期: 2021-04-13 00:48:27
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