How is $$\int_0^T\cos(2\omega t+ 2\theta) dt = 0$$ Regards
Evaluating Cosine in Integral
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integration
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0What can be said about $w$, $\theta$ and $T$? – 2012-10-31
2 Answers
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$$\frac{d}{dx}\sin(mx + c) = m cos(mx + c)$$
so backwards we have
$$\int \cos(2wt + 2 \theta) dt = \frac{1}{2w} \sin(2wt + 2 \theta) + c $$
and you can evaluate that at the limits.
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0@PeterPhipps, thanks! – 2012-10-31
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Following Sperners' solution:
$$\int_0^T\cos (2wt+2\theta)\,dt=\left.\frac{1}{2w}\sin (2wt+2\theta)\right|_0^T=\frac{1}{2w}\left[\sin\left(2wT+2\theta\right)-\sin \left(2\theta\right)\right]$$
and the above is zero iff
$$\sin(2wT+2\theta)=\sin 2\theta\Longleftrightarrow 2wT+2\theta=2\theta\,\,\vee\,\,2wT+2\theta=\pi-2\theta$$
So the claim is false unless some relations or given values apply to $\,w,T,\theta\,$