Patricia Crane

2022-01-01

Solve for x which is in $[0,2\pi ]$

$6\mathrm{cos}x+2\sqrt{2}\mathrm{sin}x=\sqrt{22}$

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?

I have solved the question by dividing both sides

Is it possible to solve it without any arc functions?

usaho4w

Beginner2022-01-02Added 39 answers

and by squaring,

We solve the quadratic equation in

Anyway, you cant

Mollie Nash

Beginner2022-01-03Added 33 answers

Yes it is possible. Write the equation as follows:

$(3+i\sqrt{2}){e}^{-ix}+(3-i\sqrt{2}){e}^{ix}=\sqrt{22}$

$x=2n\pi -i\mathrm{log}(\pm \frac{(1\pm i)\sqrt{\frac{11}{2}}}{\sqrt{2}+3i}),\text{}\text{}\text{}n\in \mathbb{Z}$

Also, you can use this well-known formula:

$\mathrm{arctan}x=\frac{12}{i}\mathrm{log}(1-ix)-\frac{12}{i}\mathrm{log}(1+ix)$

Also, you can use this well-known formula:

Vasquez

Expert2022-01-08Added 669 answers

It is possible to have approximations of the solution.

Consider that you look for the zero's of function

Since we know at least the exact trigonometric values of multiples of

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

Truncate the expansion to any order and use series reversion.

For example, use the terms up to

For

while the exact solution is

For

while the exact solution is

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