Eliminate \theta \ and \ \phi between the following eq

Marla Payton 2021-12-28 Answered
Eliminate θ and ϕ between the following equations:
{sinθ+sinϕ=xcosθ+cosϕ=ytanθ2tanϕ2=z
What Ive
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Cassandra Ramirez
Answered 2021-12-29 Author has 30 answers
By sum to product and product to sum formulas we have
{sinθ+sinϕ=2sin(θ+ϕ2)cos(θϕ2)=xcosθ+cosϕ=2cos(θ+ϕ2)cos(θϕ2)=ytanθ2tanϕ2=cos(θϕ2)cos(θ+ϕ2)cos(θϕ2)+cos(θ+ϕ2)=z{2ab=x2cb=ybcb+c=z
and since a2+c2=1 we obtain
b=±12x2+y2
c=±yx2+y2
and then
z=±12x2+y2yx2+y2±12x2+y2±yx2+y2=x2+y22yx2+y2+2y
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Joseph Lewis
Answered 2021-12-30 Author has 43 answers
Write u=tanθ2,v=tanϕ2. Then we get
{2(u+v)(1+uv)=x(1+u2)(1+v2)22u2v2=y(1+u2)(1+v2)uv=z
Now setting s=u+v and using the third relation, we obtain
{2s(1+z)=x(12z+z2+s2)2(1z2)=y(12z+z2+s2)
Dividing, we obtain the relation s=x(1z)y, which youve
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karton
Answered 2022-01-08 Author has 368 answers

Hints
Writing θ=2p,ϕ=2q
We have
sin(p+q)x=cos(p+q)y=±1x2+y2
Using this one can find cos(pq)
Now use
z1z+1==cos(p+q)cos(pq)

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