Solving a set of 3 Nonlinear Equations In the following 3 equations: k 1 </msub>

Savanah Boone 2022-07-10 Answered
Solving a set of 3 Nonlinear Equations
In the following 3 equations:
k 1 cos 2 ( θ ) + k 2 sin 2 ( θ ) = c 1
2 ( k 2 k 1 ) cos ( θ ) sin ( θ ) = c 2
k 1 sin 2 ( θ ) + k 2 cos 2 ( θ ) = c 3
c 1 , c 2 and c 3 are given, and k 1 , k 2 and θ are the unknowns. What is the best way to solve for the unknowns?
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Answers (2)

isscacabby17
Answered 2022-07-11 Author has 13 answers
Re-write in terms of double-angle expressions, and define values A, , C:
k 1 cos 2 θ + k 2 sin 2 θ = c 1 cos 2 θ = 2 c 1 k 1 k 2 k 1 k 2 =: A 2 ( k 1 k 2 ) sin θ cos θ = c 2 sin 2 θ = c 2 k 1 k 2 =: B k 1 sin 2 θ + k 2 cos 2 θ = c 3 cos 2 θ = k 1 + k 2 2 c 3 k 1 k 2 =: C
Now, A = C implies k 1 + k 2 = c 1 + c 3 (but we'd know that simply by adding your first and third equations together), so that k 2 = c 1 + c 3 k 1 and
A = c 1 c 3 2 k 1 c 1 c 3 B = c 2 2 k 1 c 1 c 3
Observe that you can already write
tan 2 θ = c 2 c 1 c 3
which gives you θ. For k 1 and k 2 , note that A 2 + B 2 = 1 implies
( c 1 c 3 ) 2 + c 2 2 = ( 2 k 1 c 1 c 3 ) 2
so that
k 1 = 1 2 ( c 1 + c 3 ± ( c 1 c 3 ) 2 + c 2 2 ) k 2 = 1 2 ( c 1 + c 3 ( c 1 c 3 ) 2 + c 2 2 )
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Ciara Mcdaniel
Answered 2022-07-12 Author has 4 answers
Hint:
k 1 cos 2 ( θ ) + k 2 sin 2 ( θ ) = c 1
k 1 sin 2 ( θ ) + k 2 cos 2 ( θ ) = c 3
c 1 + c 3 = k 1 + k 2
2 ( k 2 k 1 ) cos ( θ ) sin ( θ ) = c 2 ( k 2 k 1 ) sin 2 θ = c 2
( k 2 k 1 ) sin 2 θ + k 1 + k 2 = c 1 + c 2 + c 3
k 2 ( sin 2 θ + 1 ) + k 1 ( 1 sin 2 θ ) = c 1 + c 2 + c 3
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