Hello, I'm stuck with this derivation:
[MATH]\frac{1}{2}\left ( \cos( \theta -\gamma) +\cos (2\omega t+\theta +\gamma ) \right )=....[/MATH][MATH]\text{should become....}=\frac{1}{2}\left [ \cos( \theta -\gamma)(1+ \cos (2\omega t+2\theta)) + \sin( \theta -\gamma)\sin (2\omega t+2\theta) \right ][/MATH]
My attempt of solution:
If I start by assuming [MATH]\alpha=\omega t+\theta \quad and \quad \beta=\omega t +\gamma[/MATH]and then [MATH]\alpha -\beta=\theta -\gamma \text{ and } \alpha +\beta=2\omega t+\theta +\gamma [/MATH]
*I do it so I can use the trig forumla: [MATH]\cos (\alpha +\beta)=\cos (\alpha)\cos (\beta)-\sin(\alpha)\sin(\beta)[/MATH]
then by using the first row I have*: [MATH] \frac{1}{2}\left ( \cos( \alpha -\beta) +\cos (\alpha +\beta) \right )=\frac{1}{2}\left ( \cos( \alpha -\beta) + \cos (\alpha)\cos (\beta)-\sin(\alpha)\sin(\beta) \right )[/MATH]
But I get stuck here, I assume I am supposed to get the terms sin and for example [MATH]\sin(2\alpha)[/MATH] somehow?
Thank you
[MATH]\frac{1}{2}\left ( \cos( \theta -\gamma) +\cos (2\omega t+\theta +\gamma ) \right )=....[/MATH][MATH]\text{should become....}=\frac{1}{2}\left [ \cos( \theta -\gamma)(1+ \cos (2\omega t+2\theta)) + \sin( \theta -\gamma)\sin (2\omega t+2\theta) \right ][/MATH]
My attempt of solution:
If I start by assuming [MATH]\alpha=\omega t+\theta \quad and \quad \beta=\omega t +\gamma[/MATH]and then [MATH]\alpha -\beta=\theta -\gamma \text{ and } \alpha +\beta=2\omega t+\theta +\gamma [/MATH]
*I do it so I can use the trig forumla: [MATH]\cos (\alpha +\beta)=\cos (\alpha)\cos (\beta)-\sin(\alpha)\sin(\beta)[/MATH]
then by using the first row I have*: [MATH] \frac{1}{2}\left ( \cos( \alpha -\beta) +\cos (\alpha +\beta) \right )=\frac{1}{2}\left ( \cos( \alpha -\beta) + \cos (\alpha)\cos (\beta)-\sin(\alpha)\sin(\beta) \right )[/MATH]
But I get stuck here, I assume I am supposed to get the terms sin and for example [MATH]\sin(2\alpha)[/MATH] somehow?
Thank you