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gerryliyana

  • 3 years ago

Using Legendre function, show that:

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  1. gerryliyana
    • 3 years ago
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    Show that: \[P_{2} (\cos \theta) = \frac{ 1 }{ 4 } (1+ 3 \cos (2\theta))\]

  2. gerryliyana
    • 3 years ago
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    wanna help me again @mukushla ??

  3. gerryliyana
    • 3 years ago
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    @oldrin.bataku @ajprincess @amistre64 @abb0t wanna help me ?

  4. oldrin.bataku
    • 3 years ago
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    This is still vague... but:$$P(x)=\frac12\left(3x^2-1\right)\\P(\cos \theta)=\frac12\left(3\cos^2 \theta-1\right)$$We use a few trigonometric identities to rewrite our \(\cos^2\theta\):$$\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta\\\cos2\theta=\cos(\theta+\theta)=\cos^2\theta-\sin^2\theta\\\cos^2\theta+\sin^2\theta=1\implies\sin^2\theta=1-\cos^2\theta\\\text{so }\cos2\theta=\cos^2\theta-1+\cos^2\theta=2\cos^2\theta-1\\\text{so }\cos^2\theta=\frac{\cos2\theta+1}{2}$$Now actually use these:$$P(\cos\theta)=\frac12\left(\frac{3\cos2\theta+3}{2}-1\right)=\frac14\left(3\cos2\theta+1\right)$$

  5. gerryliyana
    • 3 years ago
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    @oldrin.bataku Cool hah.., thanks mate ;)

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