If the line x=y=z intersect the line sinA x+sinB y+sinC z=2d^2,sin2A x+sin2B y+sin2C z=d^2 then sinA/2sinB/2sinC/2 is equal to where (A+B+C=pi)

Question

Arun · Answer

i) Let x = y = z = k -------- (1)

ii) Substituting in the given other two equations from (1) and reducing, we get:
sin(A) + sin(B) + sin(C) = 2d²/k ------ (2)

and sin(2A) + sin(2B) + sin(2C) = d²/k -------- (3)

iii) Dividing (3) by (2):
[sin(2A) + sin(2B) + sin(2C)]/[sin(A) + sin(B) + sin(C)] = 1/2 ---------- (4)

iv) You may be aware from trigonometric identities for the given condition A + B + C = π, that
sin(2A) + sin(2B) + sin(2C) = 4*sin(A)*sin(B)*sin(C) and
sin(A) + sin(B) + sin(C) = 4*cos(A/2)*cos(B/2)*cos(C/2)

Substituting these in (4) above,
4*sin(A)*sin(B)*sin(C)/ 4*cos(A/2)*cos(B/2)*cos(C/2) = 1/2

==> sin(A)*sin(B)*sin(C)/cos(A/2)*cos(B/2)*c... = 1/2

==> [2sin(A/2)cos(A/2)*2sin(B/2)cos(B/2)*2si... = 1/2
{Application of multiple angle identities}

On cancellation of the corresponding terms and simplification we get,
sin(A/2)*sin(B/2)*sin(C/2) = 1/16

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If the line x=y=z intersect the line sinA x+sinB y+sinC z=2d^2,sin2A x+sin2B y+sin2C z=d^2 then sinA/2sinB/2sinC/2 is equal to where (A+B+C=pi)

If the line x=y=z intersect the line sinA x+sinB y+sinC z=2d^2,sin2A x+sin2B y+sin2C z=d^2 then sinA/2sinB/2sinC/2 is equal to where (A+B+C=pi)

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