Findthe value of (i) sin 18^(@) " " (ii) cos 18^(@) (iii) cos 36^(@)" " (iv) sin36^(@)(v) sin 72^(@)" " (vi) cos 72^(@) (vii) sin54^(@)" " (viii) cos54^(@)

Findthe value of (i) sin 18^(@) " " (ii) cos 18^(@) (iii) cos 36^(@)"

1.	Findthe value of (i) sin 18^(@) " " (ii) cos 18^(@) (iii) cos 36^(@)" " (iv) sin36^(@)(v) sin 72^(@)" " (vi) cos 72^(@) (vii) sin54^(@)" " (viii) cos54^(@)
Answer» Solution :(i) Let `theta =18^(@) ` Then `theta =18^(@)rArr 5theta =90^(@)` `rArr2theta =(90^(@) -3theta)` `rArr sin 2theta = sin (90^(@) -3theta) =cos 3 theta` `rArr2sinthetacos theta= 4 cos^(3) theta -3cos theta` `hArr 2 sin thetacos theta- 4 cos^(3) theta+ 3cos theta=0` `hArrcos theta (2 sin theta -4cos^(2) theta+ 3) =0` `hArr 2 sintheta- 4 cos^(2) theta+ 3=0[ :'cos theta= cos18^(@)ne 0]` `HARR2 sintheta- 4 (1 - sin^(2) theta) + 3 =0` `hArrsin theta= (-2 +- sqrt(4+6))/(8) =((-1 +-sqrt(5))/(4)` `hArr sin theta =((sqrt(5)-1)/(4))` `:.sin 18^(@)= ((sqrt(5)-1)/(4))` (ii) `cos^(2)18^(@) = (1- sin^(2)18^(@))` `={1-((sqrt(5)-1)^(2))/(16)} ={1-((6 -2sqrt(5))/(16)} =(10 +2sqrt(5))/(16)` `hArrcos 18^(@)=(sqrt(10 + 2sqrt(5)))/(4 )[ :'cos 18^(@) gt 0]` (iii)cos `36^(@)= (1-2 sin^(2)18^(@))` ` ={1-2 ((sqrt(5)-1)^(2)/(16)}={1-((6-2sqrt(5)))/(8)}` `=(sqrt(5)+1)/(4)` (iv) sin `36^(@)=sqrt(1- cos^(2) 36^(@))= {1 -((sqrt(5)+1)^(2))/(16)}^(1/2)` `={(10-2sqrt(5))/(16)}^(1/2) =(sqrt(10-2sqrt(5))/(4)` `(v)sin 72^(@)= sin (90^(@) -18^(@)) = cos 18^(@)= (sqrt(10+2sqrt(5)))/(4)` (vi)cos `72^(@)= cos (90^(@) -18^(@)) = sin 18^(@) = ((sqrt(5)-1))/(4)` `(vii) sin 54^(@) = sin (90^(@) -36^(@))= cos36^(@)=((sqrt(5)+1))/(4)` (VIII) ` cos 54^(@)= cos(90^(@)-36^(@))= sin36^(@) = (sqrt(10 -2sqrt(5)))/(4)`