I ham given the following problem to solve:
1.9. The program should take three numbers: a; b; c and find the roots of the quadratic equation in the form:
If the value of the determinant of the quadratic equation is negative (i.e. ∆ <0), the program should write an appropriate message.
1.10. Modify the 1.9 so that the parameters A and a are functions of the parameter ϕ, e.g.
In the program, declare functions for the above equations and save the results in an array.
I have solved the 1.9 as that was easy. But I am stuck with the following:
$f(x) = f(1 + 0.25 \sin(x)) \tag 1$
$g(x) = g(1 + 0.5 |\sin(x)|) \tag 2$
My attempt (source):
Given,
$f(x) = f(1 + 0.25 \sin(x))$
Now,
$f(1 + 0.25 \sin(x)) = \frac{a}{16} \sin^2{x} +\left (\frac{a}{2} + \frac{b}{4}\right) \sin{x} + (a+b+c)$
Let, $\sin{x} = y$
So, the above equation becomes: $y = \frac{-(2a+b) \pm \sqrt{b^2 - 4ac}} {2a}$
Hence,
$\begin{align} \sin{x} &= \dfrac{-(2a+b) \pm \sqrt{b^2 - 4ac}}{2a} \\ x &= \sin^{-1}{\left (\dfrac{-(2a+b) \pm \sqrt{b^2 - 4ac}}{2a} \right)} \end{align}$
Are this procedure and solution correct?
I think that Problem 1.10 is almost as simple as Problem 1.9. The only difference is that the parameters $a$ and $b$ or $c$ (there is no parameter $A$ in Problem 1.9, so this should be a misprint) instead of constant values are functions of the parameter $\varphi$ declared in the program (so a program solving Problem 1.10 can be a small modification of the program solving Problem 1.9). A solver has the freedom to choose the functions.
The equality $A(\phi)=A(1.0+0.25\sin(\phi))$ is not an equation to solve, it is an example of a function of a parameter $\varphi$. A bit confusing here is that $A$ from the left-hand side is a name of a function and the left-hand side equals to value of $A$ of the argument $\varphi$, whereas $A$ from the right side is a constant and the right-hand side is a product of this $A$ by $1.0+0.25\sin(\phi)$. Similarly for the other equality.