Obliczyć - z arcusem...

[tometomek91]

Oblicz:

\(\displaystyle{ \sin ( \arctg 1 + \arctg 2)}\)

[ares41]

\(\displaystyle{ \sin ( \arctg 1 + \arctg 2)=\sin(\arctg 1)\cos(\arctg 2)+\sin(\arctg 2)\cos(\arctg 1)=\\=\sin\left(\arcsin \left( \frac{1}{ \sqrt{1+1^2} } \right) \right)\cos\left(\arccos \left( \frac{1}{ \sqrt{1+2^2} } \right)\right) + \sin\left(\arcsin \left( \frac{2}{ \sqrt{1+2^2} } \right) \right)\cos\left( \arccos \left( \frac{1}{ \sqrt{1+1^2} }\right) \right)=\\= \frac{1}{ \sqrt{2} } \cdot \frac{1}{ \sqrt{5} } + \frac{2}{ \sqrt{5} } \cdot \frac{1}{ \sqrt{2} } = \frac{3}{ \sqrt{10} }}\)