[tex]\bold{Zestaw 1}\\\\Zad. 1\\\\5x^3-10x^2=0\\5x^2(x-2)=0\\5x^2=0 /:5\\x^2=0\\x=0\\\\x-2=0 /+2\\x=2\\\\\underline{\text{Odp. } x=0 \text{ v } x=2}\\\\Zad. 2\\\\x\in (0; 90)\\sinx=0.6\\sin^2x+cos^2x=1\\0.36+cos^2x=1 /-0.36\\cos^2x=0.64\\cos=0.8\\\\tgx=\frac{sinx}{cosx}\\tgx=\frac{0.6}{0.8}=\frac35*\frac54=\frac34\\\\\underline{\text{Odp. } tgx=\frac34}[/tex]
[tex]\bold{Zestaw 2}\\\\Zad. 1\\\\a=|AB|\\a=\sqrt{(2+2)^2+(2+1)^2}=\sqrt{4^2+3^2}=\sqrt{16+9}=\sqrt{25}=5\\\\h=\frac{a\sqrt3}2\\h=\frac{5\sqrt3}2\\\\\underline{\text{Odp. } h=\frac{5\sqrt3}2}[/tex]
[tex]\frac{a}{sin\alpha}=\frac{b}{sin\beta}=\frac{c}{sin\gamma}=2R\\\\\frac{6}{sin\alpha}=2*8\\\frac{6}{sin\alpha}=16 /*sin\alpha\\6=16sin\alpha /:16\\\frac6{16}=sin\alpha\\sin\alpha=\frac{3}8=0.375\\\alpha=22\\\\\frac{10}{sin\beta}=16\\10=16sin\beta /:16\\\frac{10}{16}=sin\beta\\sin\beta=\frac{5}{8}=0.625\\\beta=39\\\\\gamma=180-(39+22)=180-61=119\\sin119=sin(90+29)=cos29\\\\\frac{c}{cos29}=16\\\frac{c}{0.8746}=16 /*0.8746\\c=13.9936\\\\\underline{\text{Odp. c=13.99}}[/tex]
[tex]\bold{Zestaw 3}\\\\Zad. 1\\\\3*3^3-4*3^2+2*3-4=3^4-4*9+6-4=81-36+6-4=47\\\underline{\text{Odp. w(3)=47}}[/tex]
[tex]Zad. 2\\A=(-2, 2)\\B=(2, 10)\\S=(\frac{-2+2}2; \frac{2+10}2)\\S=(\frac02; \frac{12}2)\\S=(0, 6)\\\\\left \{ {{2=-2a+b} \atop {10=2a+b}} \right. \\12=2b /:2\\6=b\\10=2a+b /-6\\4=2a /:2\\2=a\\\\2*a_2=-1 /:2\\a_2=-\frac12\\\\6=-\frac12*0+b\\6=b\\\\\underline{\text{Odp. }y=-\frac12x+6}[/tex]
[tex]\bold{Zestaw 4}\\\\Zad. 1\\x^3-4x=0\\x(x^2-4)=0\\x=0\\x^2-4=0 /+4\\x=4\\x=2 \text{ v } x=-2\\\\\underline{\text{Odp. } x=2 \text{ v } x=2 \text{ v } x=-2}[/tex]
[tex]Zad. 2\\b - \text{ramie trojkata}\\a - \text{podstawa trojkata}\\h - \text{wysokosc trojkata}\\\alpha - \text{kat przy podstawie}\\\\h=2a\\\\(\frac12a)^2+h^2=b^2\\(\frac12a)^2+(2a)^2=b^2\\\frac14a^2+4a^2=b^2\\\frac{17}4a^2=b^2\\b=\sqrt{\frac{17}4}a=\frac{\sqrt{17}}2a\\[/tex]
[tex]\\sin\alpha=\frac{h}b\\\\sin\alpha=\frac{2a}{\frac{\sqrt{17}}2a}\\sin\alpha=2:\frac{\sqrt{17}}2\\sin\alpha=2*\frac{2}{\sqrt{17}}\\sin\alpha=\frac{4}{\sqrt{17}}\\sin\alpha=\frac{4\sqrt{17}}{17}\\\\\underline{\text{Odp. } sin\alpha=\frac{4\sqrt{17}}{17}}[/tex]
[tex]\bold{Zestaw 5}\\\\Zad. 1\\S=(2, 1)\\M=(6, 4)\\\\r=|SM|\\r=\sqrt{(6-2)^2+(4-1)^2}=\sqrt{4^2+3^2}=\sqrt{16+9}=\sqrt{25}=5\\r^2=5^2=25\\\\(x-a)^2+(y-b)^2=r^2\\\\(x-2)^2+(y-1)^2=25\\x^2-4x+4+y^2-2y+1=25\\x^2+y^2-4x-2y=25-4-1\\x^2+y^2-4x-2y=25-5\\x^2+y^2-4x-2y=20\\\\\underline{\text{Odp. } x^2+y^2-4x-2y=20}[/tex]
[tex]Zad. 2\\\alpha\in(0; 90) - \text{Wszystkie funkcje beda dodatnie}\\\\tg\alpha=2\\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\\\\tg^2\alpha=\frac{1-cos^2\alpha}{cos^2\alpha}\\[/tex]
[tex]2^2=\frac{1-cos^2\alpha}{cos^2\alpha}\\4=\frac{1-cos^2\alpha}{cos^2\alpha} /*cos^2\alpha\\4cos^2\alpha=1-cos^2\alpha /+cos^2\alpha\\5cos^2\alpha=1 /:5\\cos^2\alpha=\frac15\\cos\alpha=\sqrt{\frac15}=\frac1{\sqrt5}=\frac{\sqrt5}5\\\\\underline{\text{Odp. } cos\alpha=\frac{\sqrt5}5}[/tex]
[tex]\bold{Zestaw 6}\\\\Zad. 1\\\frac{2x-3}{4-1}=\frac34\\4(2x-3)=3*3\\8x-12=9 /+12\\8x=21 /:8\\x=\frac{21}8[/tex]
