Question

Даю 50 балів, розв'яжіть задачі в повній формі(з дано, знайти)

Answer 1

Для простоти введемо позначення:

A = CB, b= AC, c = AB, α = ∠A, β = ∠B

1. b=3, cosα = 1/4

$cosalpha =frac{b}{c} => c =frac{b}{cosalpha } \\c = 3:frac{1}{4} = 3cdot 4 = 12:: (cm)\\a= sqrt{12^2-3^3} = sqrt{144-9} = sqrt{135} approx 11,6 ::(cm)$

2. a = 5, sinα = 2/3

$sinalpha = frac{a}{c} => c=frac{a}{sinalpha} \\c=5:frac{2}{3} =frac{5cdot 3}{2} = frac{15}{2}=7,5 :: (cm) \\b=sqrt{(frac{15}{2})^2-5^2 } = sqrt{frac{225}{5}-25 } = sqrt{frac{125}{4}}=frac{5sqrt{5} }{2} approx 5,6 :: (cm)$

3. b = 8, tgβ = 3

$tgbeta =frac{b}{a} => a=frac{b}{tgbeta } \\a=frac{8}{3} approx 2,7 :: (cm)\\c=sqrt{8^2+(frac{8}{3} )^2} = sqrt{64+frac{64}{9} } = sqrt{frac{640}{9} }=frac{8sqrt{10} }{3} approx 8,4 :: (cm)$

4. c = 12, cosβ = 4/5

$cosbeta =frac{a}{c} => a=ccdot cosbeta \\a = 12cdot frac{4}{5} = frac{48}{5} = 9,6 :: (cm)\\b=sqrt{12^2-(frac{48}{5})^2 }= sqrt{144-frac{2304}{25} }= sqrt{frac{3600-2304}{25} }=sqrt{frac{1296}{25} }={frac{36}{5} = 7,2 :: (cm)$

5. b = 6, cosβ = 1/3

$sin^2beta +cos^2beta =1 => sin^2beta = 1- cos^2beta \\sin^2beta = 1-(frac{1}{3} )^2 = 1-frac{1}{9}=frac{8}{9} ;:: sinbeta = sqrt{frac{8}{9} } =frac{sqrt{8}}{3} \\sinbeta=frac{b}{c} => c = frac{b}{sinbeta } \\c = 6:frac{sqrt{8}}{3} = 6cdot frac{3}{sqrt{8}} = frac{18sqrt{8} }{sqrt{8}cdot sqrt{8}} = frac{18sqrt{8}}{8} = frac{9sqrt{8}}{4} approx 6,4 :: (cm)$

$a = sqrt{(frac{9sqrt{8}}{4} )^2-6^2}= sqrt{ frac{81cdot 8}{16} -36}= sqrt{ frac{648-576}{16}}=sqrt{ frac{72}{16}}= frac{sqrt {4}cdot sqrt {18}}{sqrt{16}}= frac{2 sqrt {18}}{4}= frac{sqrt {2}cdot sqrt {9}}{2} = frac{3sqrt {2}}{2} approx 2,1 :: (cm)$

6. c = 8, tgβ = 6/7

$1+tg^2beta = frac{1}{cos^2beta } => cos^2beta = frac{1}{1+tg^2beta } \\cos^2beta = frac{1}{1+(frac{6}{7} )^2 } = frac{1}{1+frac{36}{49} } = frac{1}{frac{85}{49} } = frac{49}{85} \ cosbeta = sqrt{frac{49}{85}} = frac{7}{sqrt{85}} \\cosbeta = frac{a}{c} => a = ccdot cosbeta \\a = 8cdot frac{7}{sqrt{85}}= frac{56}{sqrt{85}}$

$b=sqrt{(8^2-left(frac{56}{sqrt{85}}right)^2} =sqrt{64-frac{3136}{85}} =sqrt{frac{5440-3136}{85}} =sqrt{frac{2304}{85}} = frac{48}{sqrt{85}} approx 5,2 :: (cm)$