Cho \(\sin \alpha = \frac{8}{{17}},\sin \beta = \frac{{15}}{{17}}\) với \(0 < \alpha < \frac{\pi }{2},0 < \beta < \frac{\pi }{2}\). Chứng minh rằng \(\alpha + \beta = \frac{\pi }{2}\)
Ta có:
\(\begin{array}{l}
\cos \alpha = \sqrt {1 - \frac{{64}}{{289}}} = \frac{{15}}{{17}}\\
\cos \beta = \sqrt {1 - \frac{{225}}{{289}}} = \frac{8}{{17}}
\end{array}\)
Do đó :
\(\begin{array}{l}
\sin \left( {\alpha + \beta } \right) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \\
= \frac{8}{{17}}.\frac{8}{{17}} + \frac{{15}}{{17}}.\frac{{15}}{{17}} = \frac{{289}}{{289}} = 1
\end{array}\)
Vì \(0 < \alpha < \frac{\pi }{2},0 < \beta < \frac{\pi }{2}\) nên suy ra \(\alpha + \beta = \frac{\pi }{2}\)
-- Mod Toán 10