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15 Vereinfache:a) tan(α)⋅cos(α) \tan (\alpha) \cdot \cos (\alpha) tan(α)⋅cos(α)b) sin(α)tan(α) \frac{\sin (\alpha)}{\tan (\alpha)} tan(α)sin(α)c) sin3(α)+sin(α)⋅cos2(α) \sin ^{3}(\alpha)+\sin (\alpha) \cdot \cos ^{2}(\alpha) sin3(α)+sin(α)⋅cos2(α)d) 1tan(α)⋅cos(α) \frac{1}{\tan (\alpha) \cdot \cos (\alpha)} tan(α)⋅cos(α)1e) 1+cos(α)1−cos(α) \sqrt{1+\cos (\alpha)} \sqrt{1-\cos (\alpha)} 1+cos(α)1−cos(α) f) sin(α)+cos(α)tan(α) \sin (\alpha)+\frac{\cos (\alpha)}{\tan (\alpha)} sin(α)+tan(α)cos(α)g) sin4(α)−cos4(α) \sin ^{4}(\alpha)-\cos ^{4}(\alpha) sin4(α)−cos4(α)h) tan(α)sin(α)−tan(α)⋅sin(α) \frac{\tan (\alpha)}{\sin (\alpha)}-\tan (\alpha) \cdot \sin (\alpha) sin(α)tan(α)−tan(α)⋅sin(α)i) cos(α)1−sin(α)−1cos(α) \frac{\cos (\alpha)}{1-\sin (\alpha)}-\frac{1}{\cos (\alpha)} 1−sin(α)cos(α)−cos(α)1
Hallo,
Aufgabe c)
sin3(α)+sin(α)⋅cos2(α) \sin ^{3}(\alpha)+\sin (\alpha) \cdot \cos ^{2}(\alpha) sin3(α)+sin(α)⋅cos2(α)
= sin(α) (sin2(α) +cos2(α)) --------->sin2(α) +cos2(α) =1
=sin(α)
a) tan(α)⋅cos(α) \tan (\alpha) \cdot \cos (\alpha) tan(α)⋅cos(α) =sin(α)cos(α) \frac{sin(α)}{cos(α)} cos(α)sin(α)* cos(α) =sin(α)
b) sin(α)tan(α) \frac{\sin (\alpha)}{\tan (\alpha)} tan(α)sin(α) = cos(α)
g)
sin4(α)−cos4(α) \sin ^{4}(\alpha)-\cos ^{4}(\alpha) sin4(α)−cos4(α)
=(sin2(α)−cos2(α))(sin2(α)+cos2(α)) =(\sin ^{2}(\alpha)-\cos ^{2}(\alpha))( \sin ^{2}(\alpha)+\cos ^{2}(\alpha) )=(sin2(α)−cos2(α))(sin2(α)+cos2(α))
=sin2(α)−cos2(α) =\sin ^{2}(\alpha)-\cos ^{2}(\alpha)=sin2(α)−cos2(α)
=1−cos2(α)−cos2(α) =1-\cos ^{2}(\alpha)-\cos ^{2}(\alpha)=1−cos2(α)−cos2(α)
=1−2cos2(α) =1-2\cos ^{2}(\alpha)=1−2cos2(α)
e)
1+cos(α)1−cos(α) \sqrt{1+\cos (\alpha)} \sqrt{1-\cos (\alpha)} 1+cos(α)1−cos(α)
=(1+cos(α))(1−cos(α)) =\sqrt{(1+\cos (\alpha))(1-\cos (\alpha))} =(1+cos(α))(1−cos(α))
=12−cos2(α) =\sqrt{1^2-\cos^2 (\alpha)} =12−cos2(α)
=sinα=\sin\alpha=sinα
Für weitere Hilfen solltest du eigene Ansätze posten.
:-)
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