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[apcalculus] appearance of radians in calculus problems
Posted:
Oct 22, 2012 4:38 PM


NOTE: This apcalculus EDG will be closing in the next few weeks. Please sign up for the new AP Calculus Teacher Community Forum at https://apcommunity.collegeboard.org/gettingstarted and post messages there.  It has been very interesting to read what teachers say about this topic!
I tell my students that angle measure changes are in radians because the trig derivatives and integrals we memorize only work in radians.
Consider two graphs of the sine, one with the xaxis expressed in degrees, and the other in radians. Look at an average rate of change over, for example, pi/6 to pi/4 on the radian graph, or 30 to 45 degrees. The change in y is the same, but the change in the degree measures are different. The radian graph gives us a rate of change of (sqrt(2)1)/2 divided by pi/12, but the degree graph gives us (sqrt(2)1)/2 divided by 15not the same number at all!
Therefore, as we look at the instantaneous rate of change, the derivatives are different if we use degrees instead of radians. The derivative of the sine is the cosine if angles are expressed in radians, but if the angles are in degrees, the derivative of the sine is the cosine times (180/pi). Yuk. And the second derivative of the sine is negative sine times (180/pi)^2.
When I introduce the derivative of the sine function, I have the kids graph the derivative function on their calculators. Y1=sin(x), Y2=nDeriv(Y1,X,X). In the past I have made sure they are in radian mode before we start. Now, maybe I'll change that! If we try this in degree mode (even if we change the window to x = 360 to 360), the derivative doesn't resemble the value of the slope until we convert to degrees. (It IS the value of the slope, just in degree mode.)
I don't know how well I've explained this, but it seems to make sense to my students.
Ginger Riddle Leavenworth, Ks
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