solving three equation in variable form to find 3 unknown

[miko]

Hello
I have 3 equatons with 3 unknown θ2,θ3 and θ4 given below​

x = a4 Cos θ1 Cos (θ2 + θ3+ θ4 ) + a3 Cos θ1 Cos (θ2 + θ3) + a2 Cos θ1 Cos θ2
y = a4 Sin θ1 Cos (θ2 + θ3+ θ4 ) + a3 Sin θ1 Cos (θ2 + θ3) + a2 Sin θ1 Cos θ2 ​

z = a4 Sin (θ2 + θ3+ θ4 ) + a3 Sin (θ2 + θ3) + a2 Sin θ2

can you plz help me find value of θ2,θ3,θ4 in terms of x, y, z, a2, a3, a4 and θ1 (known variables)
I will be very thankful to you
Regards
Miko

 

[tkhunny]

You MAY be able to simplify your life by complicating it. I explain...

My first impression is to expand all the multiple angle trigonometry functions and transform averything to a sine or cosine of a single angle. You can then substitute these with expressions (maybe \(\displaystyle cos(\theta_{1}) = W\), etc.) that may lead to a more tractable algebraic expression.

Like I said, this is just my first impression, but it may lead to something.

 

[soroban]

Hello, miko!

What a hideous problem! . . . Where did it come from?

\(\displaystyle \begin{array}{ccc}x &=& a_4\cos\theta_1\cos(\theta_2+\theta_3+\theta_4) + a_3\cos\theta_1\cos(\theta_2+\theta_3) + a_2\cos\theta_1\cos\theta_2 \\
y &=& a_4\sin\theta_1\cos(\theta_2+\theta_3+\theta_4) + a_3\sin\theta_1\cos(\theta_2+\theta_3) + a_2\sin\theta_1\cos\theta_2 \\
z &=& a_4\sin(\theta_2+\theta_3+\theta_4) \;\;+\;\; a_3\sin(\theta_2+\theta_3) \;\;+\;\; a_2\sin\theta_2 \end{array}\)

\(\displaystyle \text{Solve for }\theta_2,\,\theta_3,\,\theta_4.\)

Even with tkhunny's suggestion, the problem will be unmanageable.
I don't think it can be solved.


Is there an \(\displaystyle a_1\) ?


Do they know how much I hate subscripts?

Imagine being choreographer to four identically-costumed and masked dancers.
And the only difference is the color of their shoes.

"You there . . . Heather? . . . Darla? (no response)
. . um, you in the red shoes ... hold your right arm higher.
And, ah, you in the blue shoes ... left leg straight."

 

[Deleted member 4993]

Hello
I have 3 equatons with 3 unknown θ2,θ3 and θ4 given below​

x = a4 Cos θ1 Cos (θ2 + θ3+ θ4 ) + a3 Cos θ1 Cos (θ2 + θ3) + a2 Cos θ1 Cos θ2
y = a4 Sin θ1 Cos (θ2 + θ3+ θ4 ) + a3 Sin θ1 Cos (θ2 + θ3) + a2 Sin θ1 Cos θ2
​

z = a4 Sin (θ2 + θ3+ θ4 ) + a3 Sin (θ2 + θ3) + a2 Sin θ2

can you plz help me find value of θ2,θ3,θ4 in terms of x, y, z, a2, a3, a4 and θ1 (known variables)
I will be very thankful to you
Regards
Miko

These equations are not independent.

y/x = Tan (θ1)

Thus those unknowns cannot be solved uniquely.