Zad. 1
[latex](sinalpha +cosalpha)^2+(sinalpha -cosalpha)^2=2[/latex]
[latex]sin^2 alpha +2sinalpha cosalpha +cos^2 alpha +sin^2 alpha -2sinalpha cosalpha +cos^2 alpha=2[/latex]
[latex]2sin^2alpha +2cos^2alpha=2[/latex]
[latex]2(sin^2alpha + cos^2alpha)=2[/latex]
[latex]2cdot 1 = 2[/latex]
[latex]2=2[/latex]
[latex]L=P[/latex]
Zad. 2.
a.
[latex]sinx=dfrac{sqrt{3}}{2}[/latex]
[latex]x=dfrac{pi}{3}+2kpi ,quad kin C[/latex]
[latex]x=dfrac{2pi}{3}+2kpi , quad kin C[/latex]
b.
[latex]ctgx=1[/latex]
[latex]x=dfrac{pi}{4}+kpi ,quad kin C[/latex]
Zad. 3.
[latex]cosalpha =frac{1}{6},quad alpha in (270^{circ};360^{circ})[/latex]
[latex]cos^2alpha + sin^2alpha=1[/latex]
[latex](frac{1}{6})^2+sin^2alpha=1[/latex]
[latex]sin^2alpha=1-frac{1}{36}[/latex]
[latex]sin^2alpha=frac{35}{36}[/latex]
[latex]sinalpha=-frac{sqrt{35}}{6}[/latex]
[latex]tgalpha =dfrac{-frac{sqrt{35}}{6}}{frac{1}{6}}=-dfrac{sqrt{35}}{6}cdot 6=-sqrt{35}[/latex]
[latex]ctgalpha =dfrac{frac{1}{6}}{-frac{sqrt{35}}{6}}=dfrac{1}{6}cdot (-dfrac{6}{sqrt{35}})= -dfrac{1}{sqrt{35}}=-frac{sqrt{35}}{35}[/latex]
Zad. 4
[latex]dfrac{sin120^{circ}}{cos150^{circ}} cdot dfrac{tg135^{circ}}{sin150^{circ}}=dfrac{sin(180^{circ}-60^{circ})}{cos(180^{circ}-30^{circ})}cdot dfrac{tg(180^{circ}-45^{circ})}{sin(180^{circ}-30^{circ})}=dfrac{sin60^{circ}}{-cos30^{circ}}cdot dfrac{-tg45^{circ}}{sin30^{circ}}=[/latex]
[latex]=dfrac{frac{sqrt{3}}{2}cdot (-1)}{-frac{sqrt{3}}{2}cdot frac{1}{2}}=dfrac{-sqrt{3}}{2}cdot (-dfrac{4}{sqrt{3}})=2[/latex]
Zad. 5
[latex](frac{sqrt{3}}{2})^2+cos^2 alpha=1[/latex]
[latex]frac{3}{4}+cos^2alpha=1[/latex]
[latex]cos^2alpha=frac{1}{4}[/latex]
[latex]cosalpha=frac{1}{2}[/latex]
[latex]cosalpha + frac{1}{sinalpha}[/latex]
[latex]dfrac{1}{2}+ dfrac{1}{frac{sqrt{3}}{2}}=dfrac{1}{2}+dfrac{2}{sqrt{3}}=dfrac{1}{2}+dfrac{2sqrt{3}}{3}=dfrac{3}{6}+dfrac{4sqrt{3}}{6}=dfrac{4sqrt{3}+3}{6}[/latex]
:)
