Question

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Answer 1

$1); ; alpha =-2pi +frac{pi}{4}=-frac{7pi}{4}\\sin(-frac{7pi}{4})=sin(-2pi +frac{pi}{4})=sin frac{pi }{4}= frac{sqrt2}{2}\\cos(-frac{7pi }{4})=cosfrac{pi}{4}= frac{sqrt2}{2}\\tg(- frac{7pi }{4})=tg frac{pi}{4}=1 \\ctg( frac{7pi}{4})=ctg frac{pi }{4}=1$

$2); ; frac{cos(-245^circ )cdot ctg135^circ }{tg189^circ } = frac{cos245^circ cdot ctg135^circ}{tg189^circ} = frac{(-)cdot (-)}{(+)}= frac{(+)}{(+)}=(+)$

$3); ; sina=0,8; ,\\ frac{pi }{2} textless a textless pi ; ; to ; ; cosa textless 0; ,; ; ctga textless 0\\cosa=-sqrt{1-sin^2a}=-sqrt{1-0,64}=-0,6\\1+ctg^2a= frac{1}{sin^2a} ; ; to ; ; ctg^2a= frac{1}{sin^2a}-1=frac{1}{0,64}-1= frac{0,36}{0,64}=(frac{0,6}{0,8})^2\\ctga=-sqrt{(frac{0,6}{0,8})^2}=-frac{0,6}{0,8}=-frac{3}{4}\\1-ctga+frac{1}{cosa}=1-(-frac{3}{4})+frac{1}{-0,6}=1+frac{3}{4}-frac{10}{6}= frac{1}{12}$

$4); ; frac{cosa+ctga}{ctga} = frac{cosa}{ctga}+ frac{ctga}{ctga}= frac{cosa}{frac{cosa}{sina}}+1=sina+1$

$5); ; sin20cdot sin40cdot sin80=sin20cdot frac{1}{2}(cos(80-40)-cos(80+40))=\\= frac{1}{2}sin20cdot (cos40-cos120)= frac{1}{2}sin20cdot (cos40-cos(90+30))=\\= frac{1}{2}cdot frac{1}{2}cdot (sin(20-40)+sin(20+40))+frac{1}{2}cdot sin20cdot sin30= \\= frac{1}{4}cdot sin(-20)+ frac{1}{4}cdot sin60+frac{1}{2}cdot sin20cdot frac{1}{2}=\\= - frac{1}{4}cdot sin20+frac{1}{4} cdot frac{sqrt3}{2}+ frac{1}{4}cdot sin20= frac{sqrt3}{8}$