I have the following trigonometric equation of 3 variables f ( &#x03B8;<!-- θ --> , &#x0

Devin Anderson

Devin Anderson

Answered question

2022-06-07

I have the following trigonometric equation of 3 variables
f ( θ , λ , ϕ ) = 3 cos ( θ ) cos ( λ ) cos ( ϕ ) ( cos ( θ ) + 3 ) sin ( λ ) sin ( ϕ )

Answer & Explanation

Samantha Reid

Samantha Reid

Beginner2022-06-08Added 22 answers

If you use the well-known half tangent substitution, you can generate a more manageable, hand-bashable quadratic equation:
tan  θ 2 = x ,   tan  ϕ 2 = y ,   tan  λ 2 = z .
Then for example, one has:
cos  θ = 1  x 2 1 + x 2  and  sin  θ = 2 x 1 + x 2 .
This saves us from having to repeatedly square numbers, and after multiplying everything out, you will obtain a quadratic in x whose discriminant is:
Δ =  16 ( y + z ) 2 ( 21 y 2 z 2 + 13 y 2 + 13 z 2 + 16 y z + 21 ) < 0
unless y + z = 0.
It took me about 20 minutes to check everything by hand, but once you have a strategy and the appropriate substitution, success is virtually assured.

Arraryeldergox2

Arraryeldergox2

Beginner2022-06-09Added 10 answers

Let x = tan θ 2 , y = λ + φ 2 , z = λ φ 2 , then g ( x , y , z ) = ( 1 x 2 ) ( 2 cos 2 y + cos 2 z + 3 ) + 4 x sin y sin z + ( 1 + x 2 ) ( cos 2 y + 2 cos 2 z 7 ) = ( 3 cos 2 y + cos 2 z 10 ) x 2 + 4 x sin y sin z + cos 2 y + 3 cos 2 z 4 = ( 6 cos 2 y 2 sin 2 z 6 ) x 2 + 4 x sin y sin z 2 sin 2 y 6 sin 2 z = 0 ,
From (1) should
(2) x = 0 , y = π j , z = π l ,
and finally
θ = 2 π k , λ = π m , φ = 2 π n λ , k , m , n Z .

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