1 = (1+sinAcosA)^2

asked by guest
on Nov 15, 2024 at 2:42 am



You asked:

Solve the equation \(1 = {\left( 1 + \sin\left( A \right) \cdot \cos\left( A \right) \right)}^{2}\) for the variable \(A\).

MathBot Answer:

The real solutions are: \[\begin{aligned}A &= \left\{\pi n\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{3.1415927 n\; \middle|\; n \in \mathbb{Z}\right\}\\A &= \left\{\pi \left(\frac{1}{2} + n\right)\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{3.1415927 n + 1.5707963\; \middle|\; n \in \mathbb{Z}\right\}\end{aligned}\]

The complex solutions are: \[\begin{aligned}A &= \left\{\frac{3 \pi}{4} + \pi n - \frac{i \log{\left(4 + \sqrt{15} \right)}}{2}\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{3.1415927 n + 2.3561945 - 1.0317185 i\; \middle|\; n \in \mathbb{Z}\right\}\\A &= \left\{\frac{3 \pi}{4} + \pi n - \frac{i \log{\left(4 - \sqrt{15} \right)}}{2}\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{3.1415927 n + 2.3561945 + 1.0317185 i\; \middle|\; n \in \mathbb{Z}\right\}\end{aligned}\]

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

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