sin(60)(sin(t)-cos(t))(sin(t)+cos(t)+2)=0

solve for t

asked by guest
on Nov 22, 2024 at 10:46 am



You asked:

Solve the equation \[\sin\left( 60 \right) \cdot \left(\sin\left( t \right) - \cos\left( t \right)\right) \cdot \left(\sin\left( t \right) + \cos\left( t \right) + 2\right) = 0\] for the unknown \(t\).

MathBot Answer:

The real solutions are: \[\begin{aligned}t &= \left\{\frac{\pi \left(5 + 8 n\right)}{4}\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{6.2831853 n + 3.9269908\; \middle|\; n \in \mathbb{Z}\right\}\\t &= \left\{\frac{\pi \left(1 + 8 n\right)}{4}\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{6.2831853 n + 0.78539816\; \middle|\; n \in \mathbb{Z}\right\}\end{aligned}\]

The complex solutions are: \[\begin{aligned}t &= \left\{\frac{5 \pi}{4} + 2 \pi n - \frac{i \log{\left(3 + 2 \sqrt{2} \right)}}{2}\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{6.2831853 n + 3.9269908 - 0.88137359 i\; \middle|\; n \in \mathbb{Z}\right\}\\t &= \left\{\frac{5 \pi}{4} + 2 \pi n - \frac{i \log{\left(3 - 2 \sqrt{2} \right)}}{2}\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{6.2831853 n + 3.9269908 + 0.88137359 i\; \middle|\; n \in \mathbb{Z}\right\}\end{aligned}\]

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

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