If in ` A B C ,A=pi/7,B=(2pi)/7,C=(4pi)/7` then `a^2+b^2+c^2` must be

If in a triangle ABC , a=5,b=4 ,A = (pi)/( 2) +B then C

In a A B C ,ifA B=x , B C=x+1,/_C=pi/3 , then the least integer value of x is 6 (b) 7 (c) 8 (d) none of these

Find the value of cos(2pi)/7+cos(4pi)/7+cos(6pi)/7

If (3-tan^2(pi/7))/(1-tan^2(pi/7))=kcos(pi/7) then the value of k is (a)1 (b) 2 (c) 3 (d) 4

In triangle A B C , if r_1=2r_2=3r_2, then a : b is equal to 5/4 (b) 4/5 (c) 7/4 (d) 4/7

In an acute angled triangle A B C ,r+r_1=r_2+r_3a n d/_B >pi/3, then b+2c<2a<2b+2c b+4c<4a<2b+4c b+4c<4a<4b+4c b+3c<3a<3b+3c

If A + B + C = (pi)/(2) , prove that cos 2A + cos 2B + cos 2C = 1 + 4 sin A sin B cos C

If A + B + C = (pi)/(2) , prove that sin 2A + sin 2B + sin 2C = 4 cos A cos B cos C

If [(cos\ (2pi)/7,-sin\ (2pi)/7),(sin\ (2pi)/7,cos\ (2pi)/7)]^k=[(1,0),(0,1)], then the least positive integral value of k , is

To find the sum sin^(2) ""(2pi)/(7) + sin^(2)""(4pi)/(7) +sin^(2)""(8pi)/(7) , we follow the following method. Put 7theta = 2npi , where n is any integer. Then " " sin 4 theta = sin( 2npi - 3theta) = - sin 3theta This means that sin theta takes the values 0, pm sin (2pi//7), pmsin(2pi//7), pm sin(4pi//7), and pm sin (8pi//7) . From Eq. (i), we now get " " 2 sin 2 theta cos 2theta = 4 sin^(3) theta - 3 sin theta or 4 sin theta cos theta (1-2 sin^(2) theta)= sin theta ( 4sin ^(2) theta -3) Rejecting the value sin theta =0 , we get " " 4 cos theta (1-2 sin^(2) theta ) = 4 sin ^(2) theta - 3 or 16 cos^(2) theta (1-2 sin^(2) theta)^(2) = ( 4sin ^(2) theta -3)^(2) or 16(1-sin^(2) theta) (1-4 sin^(2) theta + 4 sin ^(4) theta) " " = 16 sin ^(4) theta - 24 sin ^(2) theta +9 or " " 64 sin^(6) theta - 112 sin^(4) theta - 56 sin^(2) theta -7 =0 This is cubic in sin^(2) theta with the roots sin^(2)( 2pi//7), sin^(2) (4pi//7), and sin^(2)(8pi//7) . The sum of these roots is " " sin^(2)""(2pi)/(7) + sin^(2)""(4pi)/(7) + sin ^(2)""(8pi)/(7) = (112)/(64) = (7)/(4) . The value of (tan^(2)""(pi)/(7) + tan^(2)""(2pi)/(7) + tan^(2)""(3pi)/(7))xx (cot^(2)""(pi)/(7) + cot^(2)""(2pi)/(7) + cot^(2)""(3pi)/(7)) is