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We have several ways in which we can integrate the basic trigonometric functions. We can integrate the imaginary exponential representation or we can integrate the infinite-series representations, but here I want to take a different approach. I want to determine the differentials of the sine and cosine and then parlay that knowledge into the indefinite integrals that we want.

I begin by restating the addition theorems of basic trigonometry;

(Eq'n 1)


(Eq'n 2)

I then make the substitution A+B = x+dx and note that because dx is extremely small we have to an exquisitely good approximation Sindx = dx and Cosdx = 1. Now we have the differentials

(Eq'n 3)


(Eq'n 4)

Integrating those gives us

(Eq'n 5)


(Eq'n 6)

Example Application:

Solar Heating of a Plate;

    A flat plate with an area of A square meters lies horizontally on the Equator on the day of one of the Equinoxes. If we paint the plate perfectly black and ensure that no obstructions prevent sunlight from reaching it, then how much solar heat does the plate absorb in the course of that day? Assume that the flux of sunlight at the plate's location equals 3600 kilojoules per square meter per hour.

At any time t after sunrise the plate presents an area of Asin(ωt) to direct insolation, in which fact we have that ω = 180 degrees (or π radians) per twelve hours. In the interval dt the plate absorbs an amount of heat equal to

(Eq'n 7)

Over the twelve-hour elapse of the day the plate thus absorbs

(Eq'n 8)


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