Question

Find Q, I'm not sure if it is possible however.

Answer 1

9514 1404 393

Answer:

Q ≈ 0.0314148930589 radians ≈ 1.79994078613°

Explanation:

There is no algebraic solution for the set of equations with mixed polynomial and trig functions:

x·sin(Q/2) = 5

1/2x^2(Q -sin(Q)) = π/12

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However, we can use the first equation to substitute for x in the second equation. This gives ...

`\displaystyle (1)/(2)\left(\frac{5}{\sin{(Q)/(2)}}\right)^2(Q-sin(Q))=(\pi)/(12)`

This can be simplified using the half-angle identity and recast as a function of Q:

`\displaystyle f(Q)=(25(Q-sin(Q)))/(1-cos(Q))-(\pi)/(12)`

This function will have a zero at the desired value of Q. It can be solved iteratively. A graph shows an approximate value of Q is 0.314 (radians). The value can be refined by Newton's method iteration to give the angle to full calculator precision:

Q ≈ 0.0314148930589 radians ≈ 1.79994078613°

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The corresponding value of x is about 318.333447755. In short, this is a very narrow sector/segment in a relatively large circle.