My students know that if you graph the function $f(x)=A\sin (Bx+C)$ you will produce a sinusoidal wave with amplitude $A$, period length $2\pi\div B$, and a phase shift (left/right translation) of $B/C$. Using that information, they must find parameters $A, B, \text{ and } C$ to exactly match the function below. $$y=\sin x + \cos x$$ The observant student may see that $1.414\approx \sqrt 2$ and produce the correct result $y=\sqrt 2\sin(x+\frac{\pi}{4}.$ Then students try the same approach for this next function. $$y=\sin x + 2\cos x$$ Again, an observant student may see that $2.236\approx \sqrt 5$ and produce $y=\sqrt 5\sin(x+1.107).$ However, if students graph the two functions and zoom in to the tenthousandths scale, they see that the two graphs are very close but do not agree exactly. It seems that in using the number $1.107$ in our function, we have approximated to correct result but have overlooked some essential numerical relationship. We don't know where the $1.107$ came from or have an exact formula for it. Students may find it deeply unsettling to not know more about this number. When they tell me this, I tell them they have good instincts and that they should keep up the good work.
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About Me
I started this blog to share my transformation from math nerd to math nerd who loves to share math with young people. I teach high school in Philadelphia, Pennsylvania. Your comments are always welcome. Archives
September 2017
