1.(Euclid)
$$sin^2 18^\circ+sin^2 30^\circ=sin^2 36^\circ$$
or, expressing angles in radians
$$sin^2 \frac{\pi}{10}+sin^2 \frac{\pi}{6}=sin^2 \frac{\pi}{5}.$$
2.
$$cos 36^\circ -cos 72^\circ =\frac{1}{2}$$
or, expressing angles in radians
$$cos \frac{\pi}{5}-cos\frac{2\pi}{5}=\frac{1}{2}.$$
3.
i) $cos \frac{\pi}{7} -cos \frac{2\pi}{7}+cos \frac{3\pi}{7}=\frac{1}{2};$
ii) $sin \frac{\pi}{14}-sin \frac{3\pi}{14}+sin \frac{5\pi}{14}=\frac{1}{2};$
iii) $cos \frac{\pi}{7} \cdot cos \frac{2\pi}{7} \cdot cos \frac{3\pi}{7}=\frac{1}{8};$
iv) $sin \frac{\pi}{14}\cdot sin\frac{3\pi}{14} \cdot sin \frac{5\pi}{14}=\frac{1}{8}.$
4.
$$1^\circ .\;\;cos\frac{\pi}{21}-cos \frac{4\pi}{21}+cos \frac{5\pi}{21}=\frac{\sqrt{21}-1}{4};$$
$$1^{\circ \circ}.\;\;cos \frac{2\pi}{21}+cos\frac{8\pi}{21}+cos\frac{10\pi}{21}=\frac{\sqrt{21}+1}{4};$$
$$2^\circ\;\;-sin\frac{\pi}{21}+sin\frac{4\pi}{21}+sin\frac{5\pi}{21}=\frac{1}{2} \sqrt{\frac{5+\sqrt{21}}{2}};$$
$$3^\circ.\;\;sin\frac{2\pi}{21}+sin\frac{8\pi}{21}-sin\frac{10\pi}{21}=\frac{1}{2}\sqrt{\frac{5-\sqrt{21}}{2}};$$
$$4^\circ\;\;cos\frac{\pi}{21}\cdot cos\frac{4\pi}{21}\cdot cos\frac{5\pi}{21}=\frac{5+\sqrt{21}}{16};$$
$$5^\circ.\;\;cos\frac{2\pi}{21}\cdot cos\frac{8\pi}{21}\cdot cos\frac{10\pi}{21}=\frac{5-\sqrt{21}}{16};$$
$$6^\circ.\;\;sin\frac{\pi}{21}\cdot sin\frac{4\pi}{21}\cdot sin\frac{5\pi}{21}=\frac{1}{8}\sqrt{\frac{5-\sqrt{21}}{2}}.$$
Niciun comentariu:
Trimiteți un comentariu