Uzupełniona tabela:

α	π/6	π/4	π/3
sinα	1/2	√2/2	√3/2
cosα	√3/2	√2/2	1/2
tgα	√3/3	1	√3
ctgα	√3	1	√3/3

 

 

$\text{a)}\sin 2\frac{1}{3}\pi =\sin \left(2\pi +\frac{1}{3}\pi \right)=\frac{\sin 1}{3}\pi =\frac{\sin \pi }{3}=\sin {60}^{\circ }=\frac{\sqrt{3}}{2}$        

$\text{b)}{\text{tg}}_{}\frac{13}{6}\pi =\text{tg}\left(\frac{12}{6}\pi +\frac{1}{6}\pi \right)=\text{tg}\left(2\pi +\frac{1}{6}\pi \right)=\text{tg}\frac{1}{6}\pi =\text{tg}\frac{\pi }{6}=\text{tg}{30}^{\circ }=\frac{\sqrt{3}}{3}$   

$\text{c)}{\text{tg}}_{}\frac{13}{3}\pi =\text{tg}\left(\frac{12}{3}\pi +\frac{1}{3}\pi \right)=\text{tg}\left(4\pi +\frac{1}{3}\pi \right)=\text{tg}\frac{1}{3}\pi =\text{tg}\frac{\pi }{3}=\text{tg}{60}^{\circ }=\sqrt{3}$   

$\text{d)}\frac{\sin 17}{4}\pi =\sin \left(\frac{16}{4}\pi +\frac{1}{4}\pi \right)=\sin \left(4\pi +\frac{1}{4}\pi \right)=\text{sin}\frac{1}{4}\pi =\text{sin}\frac{\pi }{4}=\sin {45}^{\circ }=\frac{\sqrt{2}}{2}$   

$\text{e)}\sin \left(-4\pi \right)=\sin 0^{\circ }=0$  

$\text{f)}\cos \left(-\frac{3}{2}\pi \right)=\cos \left(-2\pi +\frac{1}{2}\pi \right)=\text{cos}\frac{1}{2}\pi =\text{cos}\frac{\pi }{2}=\cos {90}^{\circ }=0$  

$\text{g)}\cos \left(-\frac{15}{4}\pi \right)=\cos \left(-\frac{16}{4}\pi +\frac{1}{4}\pi \right)=\cos \left(-4\pi +\frac{1}{4}\pi \right)=\text{cos}\frac{1}{4}\pi =\text{cos}\frac{\pi }{4}=\cos {45}^{\circ }=\frac{\sqrt{2}}{2}$   

$\text{h)}\cos \left(-\frac{11}{3}\pi \right)=\cos \left(-\frac{12}{3}\pi +\frac{1}{3}\pi \right)=\cos \left(-4\pi +\frac{1}{3}\pi \right)=\text{cos}\frac{1}{3}\pi =\text{cos}\frac{\pi }{3}=\cos {60}^{\circ }=\frac{1}{2}$   

$\text{i)}\sin \left(-\frac{23}{4}\pi \right)=\sin \left(-\frac{24}{4}\pi +\frac{1}{4}\pi \right)=\sin \left(-6\pi +\frac{1}{4}\pi \right)=\text{sin}\frac{1}{4}\pi =\text{sin}\frac{\pi }{4}=\sin {45}^{\circ }=\frac{\sqrt{2}}{2}$