Question

For the function v = 12 sin 40, calculate the: a) mean b) root mean square (RMS) 

Over a range of 0 ≤ Ø ≤ π/4 radians. 

(Note: the trigonometric identity cos 2Ø = 1-2sin^2 Ø)

Answer 1

$$v_{av}=12\frac{4}{\pi}\int _0^{\frac{\pi}{4}} \sin{4\theta}d\theta=3\frac{4}{\pi}\int _0^{\pi} \sin{x}dx$$ $$v_{av}=-3(\cos{\pi}-\cos{0})\frac{4}{\pi}=\frac{24}{\pi}$$

$$v_{rms}^2=12^2\frac{4}{\pi}\int _0^{\frac{\pi}{4}} \sin^2{4\theta}d\theta=12^2\frac{2}{\pi}\int _0^{\frac{\pi}{4}} (1-\cos{8\theta})d\theta$$ $$\int _0^{\frac{\pi}{4}} (1-\cos{8\theta})d\theta=\frac{\pi}{4}-\frac{1}{8}\int _0^{2\pi} \sin{y}dy=\frac{\pi}{4}$$

Thus,

$$v_{rms}^2=12^2\frac{4}{\pi}\int _0^{\frac{\pi}{4}} \sin^2{4\theta}d\theta=12^2\frac{2}{\pi}\frac{\pi}{4}$$ $$v_{rms}=\frac{12}{\sqrt{2}}$$