Verified answer

[tex]\displaystyle\bf\\1)\\\\Cos1770^\circ=Cos\Big(360^\circ\cdot 4+330^\circ\Big)=Cos330^\circ=\\\\\\=Cos\Big(360^\circ-30^\circ\Big)=Cos30^\circ=\frac{\sqrt{3} }{2}\\\\2)\\\\Sin\frac{27\pi }{2} =Sin\Big(12\pi +\frac{3\pi }{2} \Big)=Sin\Big(2\pi \cdot 6+\frac{3\pi }{2} \Big)=Sin\frac{3\pi }{2} =-1\\\\3)\\\\Ctg\Big(-\frac{23\pi }{4} \Big)=-Ctg\Big(5\pi +\frac{3\pi }{4} \Big)=-Ctg\frac{3\pi }{4} =\\\\\\=-Ctg\Big(\pi -\frac{\pi }{4} \Big)=Ctg\frac{\pi }{4} =1[/tex]