Odpowiedź:
zad 1
a)
[tex]sin30 = \frac{x}{8} \\ \frac{1}{2} = \frac{x}{8} \\ x = 4 \\ cos30 = \frac{y}{8} \\ \frac{ \sqrt{3} }{2} = \frac{y}{8} \\ y = 4 \sqrt{3} [/tex]
b)
[tex]sin30 = \frac{3}{z} \\ \frac{1}{2} = \frac{3}{z} \\ z = 6 \\ tg30 = \frac{3}{x} \\ \frac{ \sqrt{3} }{3} = \frac{3}{x} \\ x = \frac{9}{ \sqrt{3} } [/tex]
c)
[tex]cos30 = \frac{5\sqrt{3} }{z} \\\frac{\sqrt{3} }{2} = \frac{5\sqrt{3} }{z}\\z = \frac{10\sqrt{3} }{\sqrt{3} }\\z = 10\\tg30 = \frac{x}{5\sqrt{3} } \\\frac{\sqrt{3} }{3} = \frac{x}{5\sqrt{3} }\\x = \frac{5*3}{3} = 5[/tex]
d)
[tex]cos30 = \frac{9}{z} \\\frac{\sqrt{3} }{2} = \frac{9}{z}\\z= \frac{18}{\sqrt{3} } \\tg30 = \frac{x}{9} \\\frac{\sqrt{3} }{3} = \frac{x}{9} \\x = 3\sqrt{3}[/tex]
zad 2:
a)
[tex]tg60 = \frac{x}{6} \\\sqrt{3} = \frac{x}{6}\\x = 6\sqrt{3} \\y= 6\\cos60 = \frac{6}{z} \\\frac{1}{2} = \frac{6}{z} \\z = 12\\P= \frac{x*y}{2} = \frac{36\sqrt{3} }{2} = 18\sqrt{3} cm^{2} \\Ob = x + y + z = 6\sqrt{3} + 6 + 12 = 18+6\sqrt{3}cm[/tex]
b)
[tex]tg60 = \frac{10\sqrt{3} }{y} \\\sqrt{3} = \frac{10\sqrt{3} }{y} \\y=10\\cos60 = \frac{10\sqrt{3} }{z} \\\frac{1}{2} = \frac{10\sqrt{3} }{z}\\z = 20\sqrt{3} \\P = \frac{10\sqrt{3} * 10 }{2} = 50\sqrt{3} cm^{2} \\Ob = 10\sqrt{3} +10+20\sqrt{3} = 10+30\sqrt{3}[/tex]
zad 3:
a = 4cm
b = 10cm
Przyprostokątna trójkąta rownoramiennego: [tex]\frac{10-4}{2} = 3[/tex]
c - ramię
h - wysokosc trapeu
[tex]tg60 = \frac{3}{h} \\\sqrt{3} = \frac{3}{h} \\h = \frac{3}{\sqrt{3} } \\[/tex]
[tex]cos60 = \frac{3}{c} \\\frac{1}{2} = \frac{3}{c}\\c = 6\\P = \frac{1}{2}(a+b)*h\\P = \frac{1}{2} * 14 * \frac{3}{\sqrt{3} } \\P= 7 * \frac{3}{\sqrt{3} }\\P = \frac{21}{\sqrt{3} } \\Ob = a+b+2c = 4+10+2*6 = 14+12=28[/tex]
b)
a = 12
c = 26
b = 12+x
h = ?
[tex]sin60 = \frac{h}{26} \\\frac{\sqrt{3} }{2} = \frac{h}{26}\\h=13\sqrt{3} \\cos60 = \frac{x}{26}\\\frac{1}{2} = \frac{x}{26}\\x = 13\\b = 12+13=26\\P = \frac{1}{2} * (a+b)*h\\P =\frac{1}{2}*(12+26)*13\sqrt{3} \\P = \frac{1}{2} * 38 * 13\sqrt{3} \\P = 19*13\sqrt{3} = 247\sqrt{3}cm^{2} \\Ob = a+b+c+h \\Ob = 12 + 12+13 +26 + 13\sqrt{3} \\Ob = 63 + 13\sqrt{3}[/tex]
