Matematyka
Pitre
2017-06-23 02:17:34
Uzasadnianie Tożsamości, kompletnie nie ogarniam ;(
Odpowiedź
Usta18
2017-06-23 09:06:55

Uzasadnić tożsamość trygonometryczną to znaczy wykazać, że lewa strona równania jest równa prawej. [latex]a)\sinalpha + cosalpha = sqrt{1+2sinalpha cosalpha} |^{2}\\(sinalpha+cosalpha)^2} = 1+2sinalphac cosalpha\\sin^{2}alpha + cos^{2}alpha = 1, zatem:\\L = (sinalpha + cosalpha)^{2} = sin^{2}alpha+ 2sinalpha cosalpha+ cos^{2}alpha = 1+2sinalpha cosalpha =P[/latex] [latex]b)\L = cosalpha - frac{1}{cosalpha} = frac{cos^{2}alpha - (sin{2}alphan + cos^{2}alpha)}{cosalpha}=frac{cos^{2}alpha - sin^{2}alpha - cos^{2}alpha}{cosalpha} = -frac{sin^{2}alpha}{cosalpha}=\\=-frac{sinalpha}{cosalpha}cdot sinalpha=-tgalpha sinalpha =P[/latex] [latex]d)\L = frac{1}{sin^{2}alpha} + frac{1}{cos^{2}alpha} = frac{cos^{2}alpha+sin^{2}alpha}{sin^{2}alpha cos^{2}alpha} = frac{1}{sin^{2}alpha cos^{2}alpha} = P[/latex] [latex]e)\L =(1-sinalpha)(1+sinalpha)=1 - sin^2}alpha = sin^{2}alpha + cos^2}alpha - sin^{2}alpha = cos^{2}alpha = P[/latex] [latex]f)\L = (sinalpha + cosalpha)^{2} + (sinalpha - cosalpha)^{2} =\\= sin^{2}alpha + 2sinalpha cosalpha + cos^{2}alpha+ sin^{2}alpha - 2sinalpha cosalpha + cos^{2}alpha=\\=2sin{2}alpha + 2cos^{2}alpha = 2(sin^{2}alpha + cos^{2}alpha) = 2cdot1 = 2 = P[/latex] [latex]g)\L = sin^{4}alpha - cos^{4}alpha =(sin^{2}alpha+cos^{2}alpha)(sin^{2}alpha - cos^{2}alpha)=\\=1cdot[sin^{2}alpha - (1-sin^{2}alpha)] =sin^{2}alpha - 1 + sin^{2}alpha = 2sin^{2}-1 = P[/latex]

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