Question

Найдите угол альфа, если:
1) cos^2α - sin^2α=1/2
2) sinαcosα=1/4

Answer 1

1)

$\cos^2{\alpha}-\sin^2{\alpha}=\frac{1}{2} \\ \\ \cos^2{\alpha}-\sin^2{\alpha}=\cos{2\alpha} \\ \\ \cos{2\alpha} =\frac{1}{2} \\ \\ 2\alpha =\pm\frac{\pi}{3}+2\pi n, \ n\in Z \\ \\ \alpha_{1,2} =\pm\frac{\pi}{6}+\pi n, \ \in Z$

2)

$\sin{\alpha}\cdot \cos{\alpha}=\frac{1}{4} \\ \\ \sin{\alpha}\cdot \cos{\alpha}=\frac{1}{2}\cdot 2 \cdot \sin{\alpha}\cdot \cos{\alpha}=\frac{1}{2}\sin{2\alpha} \\ \\ \frac{1}{2}\sin{2\alpha}=\frac{1}{4} \\ \\ \sin{2\alpha}=\frac{1}{2} \\ \\ 2\alpha_1=\frac{\pi}{6}+2\pi n, \ n \in Z; \ \ \ 2\alpha_2=\frac{5\pi}{6}+2\pi n , \ n\in Z \\ \\ \alpha_1=\frac{\pi}{12}+\pi n, \ n \in Z; \ \ \ \alpha_2=\frac{5\pi}{12}+\pi n , \ n\in Z$