cosx-cos2x=1/2

asked by guest
on Nov 16, 2024 at 2:29 am



You asked:

Solve the equation \(\cos\left( x \right) - \cos\left( 2 x \right) = \frac{1}{2}\) for the variable \(x\).

MathBot Answer:

The real solutions are: \[\begin{aligned}x &= \left\{\pi + 2 \pi n + \arctan{\left(\frac{\sqrt{5 + \sqrt{5}} \left(\sqrt{2} + \sqrt{10}\right)}{4} \right)}\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{6.2831853 n + 4.3982297\; \middle|\; n \in \mathbb{Z}\right\}\\x &= \left\{\pi - \arctan{\left(\frac{\sqrt{5 + \sqrt{5}} \left(\sqrt{2} + \sqrt{10}\right)}{4} \right)} + 2 \pi n\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{6.2831853 n + 1.8849556\; \middle|\; n \in \mathbb{Z}\right\}\\x &= \left\{2 \pi n + \arctan{\left(\frac{\sqrt{5 - \sqrt{5}} \left(\sqrt{10} - \sqrt{2}\right)}{4} \right)}\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{6.2831853 n + 0.62831853\; \middle|\; n \in \mathbb{Z}\right\}\\x &= \left\{- \arctan{\left(\frac{\sqrt{5 - \sqrt{5}} \left(\sqrt{10} - \sqrt{2}\right)}{4} \right)} + 2 \pi + 2 \pi n\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{6.2831853 n + 5.6548668\; \middle|\; n \in \mathbb{Z}\right\}\end{aligned}\]

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