Trigonometric

# Trigonometric - DERIVATIVES OF TRIGONOMETRIC FUNCTIONS If...

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DERIVATIVES OF TRIGONOMETRIC FUNCTIONS If we sketch the graph of the function f ( x ) = sin x , since f ( x ) is periodic, we can guess that f 0 ( x ) is also periodic. Furthermore, Since f 0 (0) = 1, f 0 ( π 2 ) = 0 and so on, it is natural to think that f 0 ( x ) = cos x . We can try to conﬁrm this guess. f 0 ( x ) = lim h 0 sin( x + h ) - sin x h = lim h 0 sin x cos h + cos x sin h - sin x h = lim h 0 h sin x cos h - sin x h + cos x sin h h i = sin x · lim h 0 cos h - 1 h + cos x · lim h 0 sin h h Here, we have to calculate two limits: lim h 0 sin h h and lim h 0 cos h - 1 h . We use a geometric idea to ﬁnd those limits. In Figure 1., we can see that the area of 4 ABC, ( 4 ABC), equals 1 2 sin h . Also, ( 4 ABD) = 1 2 tan h , and the area of the sector of the circle ABC is 1 2 h . Moreover, among those three areas, ( 4 ABD) is the greatest, while ( 4 ABC) is the smallest. Thus, we can get the following inequality. 1 2 sin h < 1 2 h < 1 2 tan h If we multiply it by 2 and divide it by sin h , we get 1 < h sin h < 1 cos h (Note that we are assuming h > 0 here.) By taking inverse, we have cos h <

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Trigonometric - DERIVATIVES OF TRIGONOMETRIC FUNCTIONS If...

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