Formulas needed in Section 3.4
http://www.math.wustl.edu/~freiwald/Math131/trigderivativesÞ:.0
Trig identities
sinÐ  BÑ œ  sin B
sin# B  cos# B œ "
sin(E  FÑ œ sin E cos F  cos A sin F
cosÐE  FÑ œ cos E cos F  sin E sin F
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From geometry
A leg in a right triangle is shorter than the hypotenuse.
The tangent line at a point on a circle is perpendicular to the radius.
Consider a central angle ) (measured in radians) in a circle of radius < À
The arc of the circle between E and F has length = œ <)Þ
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We will also need
and
lim sin ) œ
)Ä! )
lim cos ))  "
)Ä!
"
(which we will prove geometrically  over)
œ!
(which we will also prove)
We will show that lim sin) ) œ ".
)Ä!
In the figure, we start with a circle of radius < œ ". Draw the chord EF , draw
FG ¼ SE, and draw the tangent lines at E and F ; they are ¼ to SE and SF . Let
them intersect at I . We use symbols like lFGl to represent the length of a line segment.
But
1) lFGl  lEFl  arcÐEFÑ œ <) œ ") œ )
lFGl
sin ) œ lFGl
lSFl œ " œ lFGl,
sin )
so
sin )  ). Since )  0, dividing by ) gives us À
)  "
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2) ) œ arcÐEFÑ  lFIl  lEIl  lIHl  lEIl œ lEHl
But tan ) œ
lEHl
lSEl
œ
lEHl
"
Multiplying by
cos )
)
3) Combining 1) and 2) gives cos ) 
sin )
)
so )  tan ) œ
sin )
cos ) .
œ lEHlß
By the Squeeze Theorem,
gives us:
cos ) 
sin )
)
 ", so lim  cos )  lim sin) )  lim  "
lim sin) ) œ "Þ
)Ä!
)Ä!
)Ä!
Æ
"
Æ
"
)Ä!
A similar argument can be used to show lim sin) ) œ "Þ THEREFORE lim sin) ) œ "Þ
)Ä!
)Ä!