sin4theta - cos4theta upon sintheta - cos theta = sintheta + cos theta

asked by guest
on Sep 24, 2024 at 5:27 pm



You asked:

Solve the equation \(\sin\left( 4 θ \right) - \frac{\cos\left( 4 θ \right)}{\sin\left( θ \right)} - \cos\left( θ \right) = \sin\left( θ \right) + \cos\left( θ \right)\) for the variable \(θ\).

MathBot Answer:

The real solutions are: \[\begin{aligned}θ &= \left\{2 \arctan{\left(\operatorname{CRootOf} {\left(1 + x^{10} - 58 x^{6} - 39 x^{2} - 8 x^{7} - 7 x^{8} - 4 x^{9} + 4 x + 8 x^{3} + 166 x^{4}, 2\right)} \right)} + 2 \pi n\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{6.2831853 n + 1.8457211\; \middle|\; n \in \mathbb{Z}\right\}\\θ &= \left\{2 \arctan{\left(\operatorname{CRootOf} {\left(1 + x^{10} - 58 x^{6} - 39 x^{2} - 8 x^{7} - 7 x^{8} - 4 x^{9} + 4 x + 8 x^{3} + 166 x^{4}, 3\right)} \right)} + 2 \pi n\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{6.2831853 n + 2.7967185\; \middle|\; n \in \mathbb{Z}\right\}\\θ &= \left\{2 \pi + 2 \arctan{\left(\operatorname{CRootOf} {\left(1 + x^{10} - 58 x^{6} - 39 x^{2} - 8 x^{7} - 7 x^{8} - 4 x^{9} + 4 x + 8 x^{3} + 166 x^{4}, 0\right)} \right)} + 2 \pi n\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{6.2831853 n + 5.2538713\; \middle|\; n \in \mathbb{Z}\right\}\\θ &= \left\{2 \pi + 2 \arctan{\left(\operatorname{CRootOf} {\left(1 + x^{10} - 58 x^{6} - 39 x^{2} - 8 x^{7} - 7 x^{8} - 4 x^{9} + 4 x + 8 x^{3} + 166 x^{4}, 1\right)} \right)} + 2 \pi n\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{6.2831853 n + 6.0476624\; \middle|\; n \in \mathbb{Z}\right\}\end{aligned}\]

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