文档介绍：Section
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Differentiation of Trigonometric
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We now take up the question of differentiating the trigonometric functions. We will start
with the sine function. From Section , we know that
d sin(x + h) − sin(x)
sin(x) = lim . ()
dx h→0 h
From the addition formula for sine we have
sin(x + h) = sin(x) cos(h) + sin(h) cos(x), ()
and so () es
d sin(x) cos(h) + sin(h) cos(x) − sin(x)
sin(x) = lim . ()
dx h→0 h
Now
sin(x) cos(h) + sin(h) cos(x) − sin(x) sin(x)(cos(h) − 1) + cos(x) sin(h)
=
h h
cos(h) − 1 sin(h)
= sin(x) + cos(x) .
h h
Thus
d cos(h) − 1 sin(h)
sin(x) = sin(x) lim + cos(x) lim . ()
dx h→0 h h→0 h
Our problem es down to evaluating the two limits in (). The second of these
turns out to be the key, so we will begin with it.
π
For 0 < h < 2 , consider the point C = (cos(h), sin(h)) on the unit circle centered at
the origin. We first repeat an argument from Section to show that sin(h) < h: If we
let A = (0, 0) and B = (1, 0), as in Figure , then the area of 4ABC is
1
sin(h).
2
h
The area of the sector of the circle cut off by the arc from B to C is the fraction 2π of the
area of the entire circle; hence, this area is
h h
π= .
2π 2
1
2 Differentiation of Trigonometric F