3cos(X)-cos(2X)=0.0047

solve for X

asked by guest
on Jan 24, 2025 at 12:14 am



You asked:

Solve the equation \[3 \cos\left( X \right) - \cos\left( 2 X \right) = 0.0047\] for the unknown \(X\).

MathBot Answer:

The real solutions are: \[\begin{aligned}X &= \left\{\pi + 2 \pi n + \arctan{\left(\frac{\sqrt{-4151 + 50 \sqrt{42406}} \cdot \left(\sqrt{63609} + 75 \sqrt{6}\right)}{9953} \right)}\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{6.2831853 n + 4.4289742\; \middle|\; n \in \mathbb{Z}\right\}\\X &= \left\{\pi - \arctan{\left(\frac{\sqrt{-4151 + 50 \sqrt{42406}} \cdot \left(\sqrt{63609} + 75 \sqrt{6}\right)}{9953} \right)} + 2 \pi n\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{6.2831853 n + 1.8542111\; \middle|\; n \in \mathbb{Z}\right\}\end{aligned}\]

The complex solutions are: \[\begin{aligned}X &= \left\{i \left(- \log{\left(150 + \sqrt{42406} + \sqrt{6} \sqrt{4151 + 50 \sqrt{42406}} \right)} - 2 \pi i n + \log{\left(200 \right)}\right)\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{i \left(- 6.2831853 n i - 1.1791916\right)\; \middle|\; n \in \mathbb{Z}\right\}\\X &= \left\{i \left(- \log{\left(150 + \sqrt{42406} - \sqrt{6} \sqrt{4151 + 50 \sqrt{42406}} \right)} - 2 \pi i n + \log{\left(200 \right)}\right)\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{i \left(- 6.2831853 n i + 1.1791916\right)\; \middle|\; n \in \mathbb{Z}\right\}\end{aligned}\]

\(i\) is the imaginary unit, defined as \(i^2 = -1\).

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