How does one solve these numerically?

[goonking]

Homework Statement

These are textbook solutions to sample problems, how would one solve these numerically (they are two different problems)?
1)

and
2)

Homework Equations

The Attempt at a Solution

 

[SteamKing]

Homework Statement

These are textbook solutions to sample problems, how would one solve these numerically (they are two different problems)?
1)
View attachment 96911and
2)
View attachment 96912

Homework Equations

The Attempt at a Solution

There aren't any obvious trig identities which apply to either problem, so one is left with either making a plot of each function versus the angle θ and finding an approximate solution, or choosing different values of θ and inserting these into each equation and seeing if a solution is obtained in that manner.

 

[goonking]

There aren't any obvious trig identities which apply to either problem, so one is left with either making a plot of each function versus the angle θ and finding an approximate solution, or choosing different values of θ and inserting these into each equation and seeing if a solution is obtained in that manner.

so it is just a long process of trial and error?

 

[SteamKing]

so it is just a long process of trial and error?

Pretty much. It's one of the reasons computers were invented, because a lot of problems can be solved only by trial and error, and computers can be programmed to do this until a numerical solution is obtained within a specified tolerance.

 

[Simon Bridge]

That is what "solve numerically" means.
Ideally you want to be systematic about your trial and error, so you use the result of the last guess to make the next guess better - what have you tried?

 

[goonking]

That is what "solve numerically" means.
Ideally you want to be systematic about your trial and error, so you use the result of the last guess to make the next guess better - what have you tried?

basically what you said, I just start from a degree and go around in intervals of 15 or 30 degrees. Just like playing a game of hot or cold I guess.

 

[Simon Bridge]

You can use a computer plot or a hand sketch to work out good places to start, and what sort of results to expect.
For the second one you'd plot $y=\sin(\frac{\pi}{180} x)-(0.703)\frac{\pi}{180}x$ (because computers work in radians and you want x in degrees).
Your solution is every value of x where the plot crosses the x axis.
The trivial solution at x=0 should be obvious.

A common systematic approach is called "Newton-Raphson" or "Newton's method" ... look it up.
If this is coursework, then there should have been a discussion of how to get numerical solutions already ... though the course may be set up so your first experience is blind so you will appreciate the systems.

 

[ehild]

so it is just a long process of trial and error?

Not quite.
You might try the method of iterations. In case of the first problem, you can write the original equation in form tan(θ)=0.75 + sin(θ), and calculate approximations by the procedure tan(θ_i+1)=0.75 + sin(θ_i), that is, θ_i+1 = atan (0.75+sin (θ_i)). Assuming solution in the first quarter, θ must be greater than 45°, Try the first approximation θ₁=50° and get θ₂, then get θ₃ from θ₂ and so on.
You can apply a similar procedure for the second problem but you have to calculate the angle in rad.

 

[haruspex]

Not sure that it helps much, but writing $t=\tan(\theta/2)$ you can convert (1) to the quartic $t^4+\frac {16}3t^3=1$. At least it might be more obvious how to iterate that to converge quickly.

 

[Ray Vickson]

so it is just a long process of trial and error?

No, not really. Various methods of "controlled search" are available. For more information, use the following terms as search keywords: bisecting search, secant method, regula falsi, Newton's method, etc.

 

[NascentOxygen]

tan θ - sin θ = 0.75

For general iteration, rearrange this in as many ways as needed into the form: θ = f(θ)

There are two obvious ways to isolate a θ from the above equation, but in general you may need to look for more (sometimes deviously) before finding arrangements that converge to all the solutions you are looking for.

e.g., your equation is equivalently
sinθ / cosθ - sinθ = 0.75

sinθ (1/cosθ - 1) = 0.75

isolating the red θ leads to a third rearrangement.

Starting with an initial guess of 70° this latter arrangement shows an oscillatory convergence to the solution, whereas the earlier arrangement converged monotonically.

In the absence of a fancy calculator, to see the convergence step-by-step you can code the calculation into one cell of a spreadsheet, so that every time you click on "recalculate" it calculates the next iteration.