| I am struggling with this problem any help would be much appreciated The two signals below are sensed by a signal processor; ?1=40sin(4?) ?2=?cos(4?) The signal processor adds the signals to form a third signal, which must be described as a distinct signal in the following form; ??=50sin(4?+?) Use a compound angle identity to determine the value of A (the amplitude of ?2). Ensure that you have your calculator in Radians (RAD) mode when determining your answer. Use graphical software to plot/model the inputs and output of the signal processor. How do you think graphical methods of sine wave combination compare with analytical methods? |
Engineering Maths problem
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Do you know what is compound angle identity?
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skeeter
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[MATH]A\cos(x) + B\sin{x} = R\sin(x + \alpha)[/MATH]
[MATH]\dfrac{A}{R} \cos{x} + \dfrac{B}{R} \sin{x} = \sin(x + \alpha)[/MATH]
[MATH]\dfrac{A}{R} \cos{x} + \dfrac{B}{R} \sin{x} = \sin{x}\cos{\alpha} + \cos{x}\sin{\alpha}[/MATH]
[MATH]\dfrac{A}{R} = \sin{\alpha}[/MATH]
[MATH]\dfrac{B}{R} = \cos{\alpha}[/MATH]
[MATH]\dfrac{A^2+B^2}{R^2} = 1 \implies R^2 = A^2+B^2[/MATH]
[MATH]\alpha = \arcsin\left(\dfrac{A}{R}\right) = \arccos\left(\dfrac{B}{R}\right)[/MATH]
[MATH]\dfrac{A}{R} \cos{x} + \dfrac{B}{R} \sin{x} = \sin(x + \alpha)[/MATH]
[MATH]\dfrac{A}{R} \cos{x} + \dfrac{B}{R} \sin{x} = \sin{x}\cos{\alpha} + \cos{x}\sin{\alpha}[/MATH]
[MATH]\dfrac{A}{R} = \sin{\alpha}[/MATH]
[MATH]\dfrac{B}{R} = \cos{\alpha}[/MATH]
[MATH]\dfrac{A^2+B^2}{R^2} = 1 \implies R^2 = A^2+B^2[/MATH]
[MATH]\alpha = \arcsin\left(\dfrac{A}{R}\right) = \arccos\left(\dfrac{B}{R}\right)[/MATH]