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Let t = tan (x/2). 1a.)Transform 2cos x - 3sin x = 7 into the form at^2 + bt + c = 0. 1b.)IfA and B are two distinct real roots of 2cos x - 3sin x = 7 . Making use of th result in (a), find the values of (i.) tan( (A+B) / 2) (ii.) (sec( (A+B) / 2))^2 (iii.) sin(A+B) 請列式作答

最佳解答:

Let t = tan (x/2). 1a .)Transform 2cos x －3sin x = 7 into the form at^2 + bt+ c = 0. Sol cosx=cos^2(x/2)－sin^2(x/2)=[cos^2(x/2)－sin^2(x/2)]/[cos^2(x/2)+sin^2(x/2)] =(1－t^2)/(1+t^2) sinx=2cos(x/2)sin(x/2)=[2cos(x/2)sin(x/2)]/[cos^2(x/2)+sin^2(x/2)] =2t/(1+t^2) 2cos x－3sin x = 7 2(1－t^2)/(1+t^2)－3*2t/(1+t^2)=7 2－2t^2－6t=7+7t^2 9t^2+6t+5=0 1b.)IfA and B are two distinct real roots of 2cos x－3sin x = 7 . Making use of th result in (a), find the values of (i.) tan( (A+B) / 2) tan(A/2)，tan(B/2) are twodistinct real roots of 9t^2+6t+5=0 tan(A/2)+tan(B/2)=6/9=－2/3 tan(A/2)*tan(B/2)=5/9 tan((A+B)/2)=(tan(A/2)+tan(B/2)\)/(1-tan(A/2)tan(B/2)) =(－2/3)[1－5/9] =(－6)/(9－5) =3/2 (ii.) (sec( (A+B) / 2))^2 =(tan( (A+B) / 2))^2+1 =9/4+1 =13/4 (iii.) sin(A+B) =2*(3/2)/[1+9/4] =3/[1+9/4] =12/13

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