Question

Using Legendre function, show that:

Answer 1

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

Answer 2

wanna help me again @mukushla ??

Answer 3

@oldrin.bataku @ajprincess @amistre64 @abb0t wanna help me ?

Answer 4

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:$$P(\cos\theta)=\frac12\left(\frac{3\cos2\theta+3}{2}-1\right)=\frac14\left(3\cos2\theta+1\right)$$

Answer 5

@oldrin.bataku Cool hah.., thanks mate ;)