Trygonometria
Podaj odpowiednie założenia i wykaż, że dla każdej wartości kąta α tożsamością jest równość:
a)
b)
c)
Założenia umiem podać sama, ale chodzi mi o to, żeby wykazać, że równość jest tożsamością :) Z góry dziękuję za pomoc
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najprościej będzie, gdy otworzysz sobie tablice matematyczne i porównasz moje przekształcenia z zawartymi tam wzorami.
a)![(2sin^{2}\alpha-1)(2sin^{2}\beta-1) = cos^{2}(\alpha+\beta)-sin^{2}(\alpha-\beta)\\ \\ L=(2sin^{2} \alpha -1)(2sin^{2} \beta -1)=(-(1-2sin^{2} \alpha))(-(1-2sin^{2} \beta))=(-cos2\alpha)(-cos2\beta)=cos2\alpha \cdot cos2 \beta\\ \\P=cos^{2}(\alpha+\beta)-sin^{2}(\alpha-\beta)=(cos\alpha cos\beta-sin\alpha sin\beta)^{2}-(sin\alpha cos\beta-cos\alpha sin\beta)^{2}=cos^{2}\alpha cos^{2}\beta-2cos\alpha cos\beta sin\alpha sin\beta+sin^{2}\alpha sin^{2}\beta-sin^{2}\alpha cos^{2}\beta+2cos\alpha cos\beta sin\alpha sin\beta -cos^{2}\alpha sin^{2}\beta=cos^{2}\alpha(cos^{2}\beta-sin^{2}\beta)-sin^{2}\alpha(cos^{2}\beta-sin^{2}\beta)=(cos^{2}\alpha-sin^{2}\alpha)(cos^{2}\beta-sin^{2}\beta)=cos2\alpha \cdot cos2\beta \\ \\L=P (2sin^{2}\alpha-1)(2sin^{2}\beta-1) = cos^{2}(\alpha+\beta)-sin^{2}(\alpha-\beta)\\ \\ L=(2sin^{2} \alpha -1)(2sin^{2} \beta -1)=(-(1-2sin^{2} \alpha))(-(1-2sin^{2} \beta))=(-cos2\alpha)(-cos2\beta)=cos2\alpha \cdot cos2 \beta\\ \\P=cos^{2}(\alpha+\beta)-sin^{2}(\alpha-\beta)=(cos\alpha cos\beta-sin\alpha sin\beta)^{2}-(sin\alpha cos\beta-cos\alpha sin\beta)^{2}=cos^{2}\alpha cos^{2}\beta-2cos\alpha cos\beta sin\alpha sin\beta+sin^{2}\alpha sin^{2}\beta-sin^{2}\alpha cos^{2}\beta+2cos\alpha cos\beta sin\alpha sin\beta -cos^{2}\alpha sin^{2}\beta=cos^{2}\alpha(cos^{2}\beta-sin^{2}\beta)-sin^{2}\alpha(cos^{2}\beta-sin^{2}\beta)=(cos^{2}\alpha-sin^{2}\alpha)(cos^{2}\beta-sin^{2}\beta)=cos2\alpha \cdot cos2\beta \\ \\L=P](https://tex.z-dn.net/?f=%282sin%5E%7B2%7D%5Calpha-1%29%282sin%5E%7B2%7D%5Cbeta-1%29+%3D+cos%5E%7B2%7D%28%5Calpha%2B%5Cbeta%29-sin%5E%7B2%7D%28%5Calpha-%5Cbeta%29%5C%5C+%5C%5C+L%3D%282sin%5E%7B2%7D+%5Calpha+-1%29%282sin%5E%7B2%7D+%5Cbeta+-1%29%3D%28-%281-2sin%5E%7B2%7D+%5Calpha%29%29%28-%281-2sin%5E%7B2%7D+%5Cbeta%29%29%3D%28-cos2%5Calpha%29%28-cos2%5Cbeta%29%3Dcos2%5Calpha+%5Ccdot+cos2+%5Cbeta%5C%5C+%5C%5CP%3Dcos%5E%7B2%7D%28%5Calpha%2B%5Cbeta%29-sin%5E%7B2%7D%28%5Calpha-%5Cbeta%29%3D%28cos%5Calpha+cos%5Cbeta-sin%5Calpha+sin%5Cbeta%29%5E%7B2%7D-%28sin%5Calpha+cos%5Cbeta-cos%5Calpha+sin%5Cbeta%29%5E%7B2%7D%3Dcos%5E%7B2%7D%5Calpha+cos%5E%7B2%7D%5Cbeta-2cos%5Calpha+cos%5Cbeta+sin%5Calpha+sin%5Cbeta%2Bsin%5E%7B2%7D%5Calpha+sin%5E%7B2%7D%5Cbeta-sin%5E%7B2%7D%5Calpha+cos%5E%7B2%7D%5Cbeta%2B2cos%5Calpha+cos%5Cbeta+sin%5Calpha+sin%5Cbeta+-cos%5E%7B2%7D%5Calpha+sin%5E%7B2%7D%5Cbeta%3Dcos%5E%7B2%7D%5Calpha%28cos%5E%7B2%7D%5Cbeta-sin%5E%7B2%7D%5Cbeta%29-sin%5E%7B2%7D%5Calpha%28cos%5E%7B2%7D%5Cbeta-sin%5E%7B2%7D%5Cbeta%29%3D%28cos%5E%7B2%7D%5Calpha-sin%5E%7B2%7D%5Calpha%29%28cos%5E%7B2%7D%5Cbeta-sin%5E%7B2%7D%5Cbeta%29%3Dcos2%5Calpha+%5Ccdot+cos2%5Cbeta+%5C%5C+%5C%5CL%3DP)
b)![\frac{1+sin2\alpha}{cos2\alpha}=\frac{1+tg\alpha}{1-tg\alpha} \\ \\ L=\frac{1+sin2\alpha}{cos2\alpha}=\frac{1+2sin\alpha cos\alpha}{cos2\alpha}=\frac{sin\alpha +cos\alpha}{(cos\alpha-sin\alpha)}=\frac{sin^{2}\alpha+cos^{2}\alpha+2sin\alpha cos\alpha}{cos^{2}\alpha-sin^{2}\alpha }=\frac{(sin\alpha+cos\alpha)^{2}}{(cos\alpha-sin\alpha)(cos\alpha+sin\alpha)}\\ \\P=\frac{1+tg\alpha}{1-tg\alpha}=\frac{1+\frac{sin\alpha}{cos\alpha}}{1-\frac{sin\alpha}{cos\alpha}}=\frac{cos\alpha + sin\alpha}{cos\alpha} \cdot \frac{cos\alpha}{cos\alpha-sin\alpha}=\frac{sin\alpha +cos\alpha}{(cos\alpha-sin\alpha)} \\ \\ L=P \frac{1+sin2\alpha}{cos2\alpha}=\frac{1+tg\alpha}{1-tg\alpha} \\ \\ L=\frac{1+sin2\alpha}{cos2\alpha}=\frac{1+2sin\alpha cos\alpha}{cos2\alpha}=\frac{sin\alpha +cos\alpha}{(cos\alpha-sin\alpha)}=\frac{sin^{2}\alpha+cos^{2}\alpha+2sin\alpha cos\alpha}{cos^{2}\alpha-sin^{2}\alpha }=\frac{(sin\alpha+cos\alpha)^{2}}{(cos\alpha-sin\alpha)(cos\alpha+sin\alpha)}\\ \\P=\frac{1+tg\alpha}{1-tg\alpha}=\frac{1+\frac{sin\alpha}{cos\alpha}}{1-\frac{sin\alpha}{cos\alpha}}=\frac{cos\alpha + sin\alpha}{cos\alpha} \cdot \frac{cos\alpha}{cos\alpha-sin\alpha}=\frac{sin\alpha +cos\alpha}{(cos\alpha-sin\alpha)} \\ \\ L=P](https://tex.z-dn.net/?f=%5Cfrac%7B1%2Bsin2%5Calpha%7D%7Bcos2%5Calpha%7D%3D%5Cfrac%7B1%2Btg%5Calpha%7D%7B1-tg%5Calpha%7D+%5C%5C+%5C%5C+L%3D%5Cfrac%7B1%2Bsin2%5Calpha%7D%7Bcos2%5Calpha%7D%3D%5Cfrac%7B1%2B2sin%5Calpha+cos%5Calpha%7D%7Bcos2%5Calpha%7D%3D%5Cfrac%7Bsin%5Calpha+%2Bcos%5Calpha%7D%7B%28cos%5Calpha-sin%5Calpha%29%7D%3D%5Cfrac%7Bsin%5E%7B2%7D%5Calpha%2Bcos%5E%7B2%7D%5Calpha%2B2sin%5Calpha+cos%5Calpha%7D%7Bcos%5E%7B2%7D%5Calpha-sin%5E%7B2%7D%5Calpha+%7D%3D%5Cfrac%7B%28sin%5Calpha%2Bcos%5Calpha%29%5E%7B2%7D%7D%7B%28cos%5Calpha-sin%5Calpha%29%28cos%5Calpha%2Bsin%5Calpha%29%7D%5C%5C+%5C%5CP%3D%5Cfrac%7B1%2Btg%5Calpha%7D%7B1-tg%5Calpha%7D%3D%5Cfrac%7B1%2B%5Cfrac%7Bsin%5Calpha%7D%7Bcos%5Calpha%7D%7D%7B1-%5Cfrac%7Bsin%5Calpha%7D%7Bcos%5Calpha%7D%7D%3D%5Cfrac%7Bcos%5Calpha+%2B+sin%5Calpha%7D%7Bcos%5Calpha%7D+%5Ccdot+%5Cfrac%7Bcos%5Calpha%7D%7Bcos%5Calpha-sin%5Calpha%7D%3D%5Cfrac%7Bsin%5Calpha+%2Bcos%5Calpha%7D%7B%28cos%5Calpha-sin%5Calpha%29%7D+%5C%5C+%5C%5C+L%3DP)
c)![\frac{cos(\alpha+\beta)}{cos(\alpha-\beta)}=\frac{1-tg\alpha tg\beta}{1+tg\alpha tg\beta} \\ \\L=\frac{cos(\alpha+\beta)}{cos(\alpha-\beta)}=\frac{cos\alpha cos\beta -sin\alpha sin\beta}{cos\alpha cos\beta +sin\alpha sin\beta} \\ \\P=\frac{1-tg\alpha tg\beta}{1+tg\alpha tg\beta}=\frac{1-\frac{sin\alpha sin\beta}{cos\alpha cos\beta}}{1+\frac{sin\alpha sin\beta}{cos\alpha cos\beta}}=\frac{cos\alpha cos\beta-sin\alpha sin\beta}{cos\alpha cos\beta}\cdot\frac{cos\alpha cos\beta}{cos\alpha cos\beta+sin\alpha sin\beta}=\frac{cos\alpha cos\beta -sin\alpha sin\beta}{cos\alpha cos\beta +sin\alpha sin\beta} \\ \\L=P \frac{cos(\alpha+\beta)}{cos(\alpha-\beta)}=\frac{1-tg\alpha tg\beta}{1+tg\alpha tg\beta} \\ \\L=\frac{cos(\alpha+\beta)}{cos(\alpha-\beta)}=\frac{cos\alpha cos\beta -sin\alpha sin\beta}{cos\alpha cos\beta +sin\alpha sin\beta} \\ \\P=\frac{1-tg\alpha tg\beta}{1+tg\alpha tg\beta}=\frac{1-\frac{sin\alpha sin\beta}{cos\alpha cos\beta}}{1+\frac{sin\alpha sin\beta}{cos\alpha cos\beta}}=\frac{cos\alpha cos\beta-sin\alpha sin\beta}{cos\alpha cos\beta}\cdot\frac{cos\alpha cos\beta}{cos\alpha cos\beta+sin\alpha sin\beta}=\frac{cos\alpha cos\beta -sin\alpha sin\beta}{cos\alpha cos\beta +sin\alpha sin\beta} \\ \\L=P](https://tex.z-dn.net/?f=%5Cfrac%7Bcos%28%5Calpha%2B%5Cbeta%29%7D%7Bcos%28%5Calpha-%5Cbeta%29%7D%3D%5Cfrac%7B1-tg%5Calpha+tg%5Cbeta%7D%7B1%2Btg%5Calpha+tg%5Cbeta%7D+%5C%5C+%5C%5CL%3D%5Cfrac%7Bcos%28%5Calpha%2B%5Cbeta%29%7D%7Bcos%28%5Calpha-%5Cbeta%29%7D%3D%5Cfrac%7Bcos%5Calpha+cos%5Cbeta+-sin%5Calpha+sin%5Cbeta%7D%7Bcos%5Calpha+cos%5Cbeta+%2Bsin%5Calpha+sin%5Cbeta%7D+%5C%5C+%5C%5CP%3D%5Cfrac%7B1-tg%5Calpha+tg%5Cbeta%7D%7B1%2Btg%5Calpha+tg%5Cbeta%7D%3D%5Cfrac%7B1-%5Cfrac%7Bsin%5Calpha+sin%5Cbeta%7D%7Bcos%5Calpha+cos%5Cbeta%7D%7D%7B1%2B%5Cfrac%7Bsin%5Calpha+sin%5Cbeta%7D%7Bcos%5Calpha+cos%5Cbeta%7D%7D%3D%5Cfrac%7Bcos%5Calpha+cos%5Cbeta-sin%5Calpha+sin%5Cbeta%7D%7Bcos%5Calpha+cos%5Cbeta%7D%5Ccdot%5Cfrac%7Bcos%5Calpha+cos%5Cbeta%7D%7Bcos%5Calpha+cos%5Cbeta%2Bsin%5Calpha+sin%5Cbeta%7D%3D%5Cfrac%7Bcos%5Calpha+cos%5Cbeta+-sin%5Calpha+sin%5Cbeta%7D%7Bcos%5Calpha+cos%5Cbeta+%2Bsin%5Calpha+sin%5Cbeta%7D+%5C%5C+%5C%5CL%3DP)
nie wiem, czy cokolwiek z tego jest czytelne, bo mi coś LaTeX świruje :/ jakby co, to pisz.