1.
[tex]3^2+21^2=d^2\\9+441=d^2\\450=d^2\\\sqrt{450}=d\\15\sqrt2=d[/tex]
[tex]d=a\sqrt2\\a\sqrt2=15\sqrt2\\a=15\\\\Ob_k=4*15=60\\Ob_p=2*3+2*21=6+42=48[/tex]
[tex]Ob_k-Ob_p=60-48=12[/tex]
Odp. Kwadrat ma obwod wiekszy o 12cm
2.
[tex]a=4cm\\c=4\sqrt{17}cm\\\\sin\alpha=\frac{b}c\\tg\alpha=\frac{b}a\\\\a^2+b^2=c^2\\16cm^2+b^2=272cm^2\\b^2=256cm^2\\b=16cm\\\\sin\alpha=\frac{16cm}{4\sqrt{17cm}}=\frac{4}{\sqrt{17}}=\frac{4\sqrt{17}}{17}\\\\tg\alpha=\frac{16cm}{4cm}=4[/tex]
3.
Wysokosc dzieli kat pomiedzy ramionami trojkata na pol, a takze podstawe trojkata na pol.
[tex]\frac12\alpha=\frac12*42=21\\\frac12d=\frac12*1,6m=0,8m[/tex]
[tex]tg(21)=\frac{\frac12d}h\\0,3839=\frac{0,8m}h /*h\\0,3839h=0,8m /:0,3839\\h=2,08m=2,1m[/tex]
Odp. Najwyzszy punkt tej drabiny znajduje sie na wysokosci 2,1m nad podloga
4.
[tex]a=12cm\\Ob=54cm\\\\Ob=a+2b\\54cm=12cm+2b\\42cm=2b /:2\\21cm=b[/tex]
[tex]6^2+h^2=21^2\\36+h^2=441\\h^2=405\\h=\sqrt{405}=9\sqrt{5}[/tex]
Kat miedzy ramionami trojkata: [tex]\alpha[/tex]
[tex]\alpha=2\gamma[/tex]
Kat przy podstawie: [tex]\beta[/tex]
[tex]sin\gamma=\frac{6}{21}=0.2857\\\gamma=17\text{ stopni}\\\alpha=2*17=34 \text{ stopnie}[/tex]
[tex]180=\alpha+2\beta\\180=34+2\beta\\146=2\beta /:2\\73=\beta[/tex]
Odp. Trojkat ma podstawe o dlugosci 12cm oraz ramiona o dlugosci 21cm. Kat miedzy ramionami tego trojkata ma miare 34 stopnie, a katy przy podstawie maja miare 73 stopnie
5.
a)
[tex]sin\alpha=\frac{15}{17}\\sin^2\alpha+cos^2\alpha=1\\(\frac{15}{17})^2+cos^2\alpha=1\\cos^2\alpha=1-\frac{225}{289}\\cos^2\alpha=\frac{64}{289}\\cos\alpha=\frac{8}{17}\\\\tg\alpha=\frac{sin\alpha}{cos\alpha}\\tg\alpha=\frac{15}{17}*\frac{17}8=\frac{15}8\\\\tg\alpha*ctg\alpha=1\\\frac{15}8*ctg\alpha=1 /*\frac8{15}\\ctg\alpha=\frac8{15}[/tex]
b)
[tex]tg\alpha=\frac5{12}\\tg\alpha=\frac{sin\alpha}{cos\alpha}\\tg^2\alpha=\frac{sin^2\alpha}{cos^2\alpha}\\\\sin^2\alpha+cos^2\alpha=1\\sin^2\alpha=1-cos^2\alpha\\\\(\frac5{12})^2=\frac{1-cos^2\alpha}{cos^2\alpha}\\\frac{25}{144}=\frac{1-cos^2\alpha}{cos^2\alpha}\\\\25cos^2\alpha=144(1-cos^2\alpha)\\25cos^2\alpha=144-144cos^2\alpha\\25cos^2\alpha+144cos^2\alpha=144\\169cos^2\alpha=144 /:169\\cos^2\alpha=\frac{144}{169}\\cos\alpha=\frac{12}{13}\\[/tex]
[tex]sin^2\alpha=1-\frac{144}{169}\\sin^2\alpha=\frac{25}{169}\\sin\alpha=\frac{5}{13}[/tex]
[tex]tg\alpha*ctg\alpha=1\\\frac5{12}*ctg\alpha=1 /*\frac{12}5\\ctg\alpha=\frac{12}5[/tex]
6.
a)
[tex]P=(3, 13) = (x, y)\\r^2=x^2+y^2\\r^2=9+169\\r^2=178\\r=\sqrt{178}\\\\sin\alpha=\frac{y}r\\sin\alpha=\frac{13}{\sqrt{178}}=\frac{13\sqrt{178}}{178}\\cos\alpha=\frac{x}r=\frac{3}{\sqrt{178}}=\frac{3\sqrt{178}}{178}\\tg\alpha=\frac{y}x=\frac{13}3\\ctg\alpha=\frac{x}y=\frac{3}{13}[/tex]
b)
[tex]P=(\sqrt{13}, \sqrt3)=(x, y)\\r^2=x^2+y^2\\r^2=13+3\\r^2=16\\r=4\\\\sin\alpha=\frac{y}r=\frac{\sqrt3}4\\cos\alpha=\frac{x}r=\frac{\sqrt{13}}4\\tg\alpha=\frac{y}x=\frac{\sqrt3}{\sqrt{13}}=\frac{\sqrt{39}}{13}\\ctg\alpha=\frac{x}y=\frac{\sqrt{13}}{\sqrt3}=\frac{\sqrt{39}}3[/tex]
7.
[tex]p=tg135=tg(90+45)=-ctg45=-1\\q=sin120=sin(90+30)=cos30=\frac{\sqrt3}2\\r=cos150=cos(180-30)=-cos30=-\frac{\sqrt3}2\\\\\text{Kolejnosc rosnaca: } p, r, q[/tex]
8.
[tex]P=\frac{ah}2=32\\\alpha=30\\\beta=75\\\\ah=64 /:h\\a=\frac{64}h\\\frac12a=\frac12*\frac{64}h=\frac{32}h\\\\tg\beta=tg75=\frac{h}{\frac12a}\\tg75=h:\frac{32}h=h*\frac{h}{21}=\frac{h^2}{32}\\\\3.7321=\frac{h^2}{32} /*32\\119.4272=h^2\\h=\sqrt{119.4272}=10.9282\\\\sin\beta=sin(75)=\frac{h}c\\0.9659=\frac{10.9282}c /*c\\0.9659c=10.9282 /:0.9659\\c=11.3140=11.31[/tex]
Odp. Ramiona tego trojkata maja miare 11.31
