Patricia Crane

2022-01-01

Solve for x which is in $\left[0,2\pi \right]$
$6\mathrm{cos}x+2\sqrt{2}\mathrm{sin}x=\sqrt{22}$
I have solved the question by dividing both sides $\sqrt{44}$, and got the answer that involves arcsin function. My question is:
Is it possible to solve it without any arc functions?

usaho4w

$6\mathrm{cos}x=\sqrt{22}-2\sqrt{2}\mathrm{sin}x$
and by squaring,
$36\left(1-{\mathrm{sin}}^{2}x\right)=8{\mathrm{sin}}^{2}x-8\sqrt{11}\mathrm{sin}x+22$
We solve the quadratic equation in $\mathrm{sin}x$ and get the two solutions
$\mathrm{sin}x=\frac{4\sqrt{11}±24\sqrt{2}}{326}$
$\mathrm{cos}x=\frac{\sqrt{22}-2\sqrt{2}\mathrm{sin}x}{6}$
Anyway, you cant

Mollie Nash

Yes it is possible. Write the equation as follows:
$\left(3+i\sqrt{2}\right){e}^{-ix}+\left(3-i\sqrt{2}\right){e}^{ix}=\sqrt{22}$

Also, you can use this well-known formula:
$\mathrm{arctan}x=\frac{12}{i}\mathrm{log}\left(1-ix\right)-\frac{12}{i}\mathrm{log}\left(1+ix\right)$

Vasquez

It is possible to have approximations of the solution.
Consider that you look for the zero's of function
$f\left(x\right)=6\mathrm{cos}x+2\sqrt{2}\mathrm{sin}x-\sqrt{22}$
Since we know at least the exact trigonometric values of multiples of $\frac{\pi }{24}$, in the considered range, it is easy to show that
$\frac{9\pi }{24}<{x}_{1}<\frac{5\pi }{12}$ and $\frac{45\pi }{24}<{x}_{2}<\frac{23\pi }{12}$
If you do not want to use a purely numerical method such as Newton which would work like a charm; consider the infinite series representation
$f\left(x\right)=6\mathrm{cos}\left(a\right)+2\sqrt{2}\mathrm{sin}\left(a\right)-\sqrt{22}+\sum _{n=1}^{\mathrm{\infty }}\frac{6\mathrm{cos}\left(a+n\frac{\pi }{2}\right)+2\sqrt{2}\mathrm{sin}\left(a+n\frac{\pi }{2}\right)}{n!}\left(x-a{\right)}^{n}$
Truncate the expansion to any order and use series reversion.
For example, use the terms up to $\left(x-a{\right)}^{6}$
For $a=\frac{9\pi }{24}$ you will obtain an explicit reult (too long to be written here) and its numerical evaluation gives
${x}_{1}=1.2259088208$
while the exact solution is ${x}_{1}=1.2259088264$
For $a=\frac{45\pi }{24}$, you will obtain
${x}_{2}=5.9382978047$
while the exact solution is ${x}_{2}=5.9382978068$

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