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15 Vereinfache:
a) tan(α)cos(α) \tan (\alpha) \cdot \cos (\alpha)
b) sin(α)tan(α) \frac{\sin (\alpha)}{\tan (\alpha)}
c) sin3(α)+sin(α)cos2(α) \sin ^{3}(\alpha)+\sin (\alpha) \cdot \cos ^{2}(\alpha)
d) 1tan(α)cos(α) \frac{1}{\tan (\alpha) \cdot \cos (\alpha)}
e) 1+cos(α)1cos(α) \sqrt{1+\cos (\alpha)} \sqrt{1-\cos (\alpha)} f) sin(α)+cos(α)tan(α) \sin (\alpha)+\frac{\cos (\alpha)}{\tan (\alpha)}
g) sin4(α)cos4(α) \sin ^{4}(\alpha)-\cos ^{4}(\alpha)
h) tan(α)sin(α)tan(α)sin(α) \frac{\tan (\alpha)}{\sin (\alpha)}-\tan (\alpha) \cdot \sin (\alpha)
i) cos(α)1sin(α)1cos(α) \frac{\cos (\alpha)}{1-\sin (\alpha)}-\frac{1}{\cos (\alpha)}

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3 Antworten

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Beste Antwort

Hallo,

Aufgabe c)

sin3(α)+sin(α)cos2(α) \sin ^{3}(\alpha)+\sin (\alpha) \cdot \cos ^{2}(\alpha)

= sin(α) (sin2(α) +cos2(α))  --------->sin2(α) +cos2(α) =1

=sin(α)

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a)  tan(α)cos(α) \tan (\alpha) \cdot \cos (\alpha) =sin(α)cos(α) \frac{sin(α)}{cos(α)} * cos(α) =sin(α)

b)  sin(α)tan(α) \frac{\sin (\alpha)}{\tan (\alpha)} = cos(α)

Avatar von 42 k
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g)

sin4(α)cos4(α) \sin ^{4}(\alpha)-\cos ^{4}(\alpha)

=(sin2(α)cos2(α))(sin2(α)+cos2(α)) =(\sin ^{2}(\alpha)-\cos ^{2}(\alpha))( \sin ^{2}(\alpha)+\cos ^{2}(\alpha) )

=sin2(α)cos2(α) =\sin ^{2}(\alpha)-\cos ^{2}(\alpha)

=1cos2(α)cos2(α) =1-\cos ^{2}(\alpha)-\cos ^{2}(\alpha)

=12cos2(α) =1-2\cos ^{2}(\alpha)


e)

 1+cos(α)1cos(α) \sqrt{1+\cos (\alpha)} \sqrt{1-\cos (\alpha)}

 =(1+cos(α))(1cos(α)) =\sqrt{(1+\cos (\alpha))(1-\cos (\alpha))}

 =12cos2(α) =\sqrt{1^2-\cos^2 (\alpha)}

=sinα=\sin\alpha

Für weitere Hilfen solltest du eigene Ansätze posten.

:-)

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