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## DIFF CALC THRY

by: Joanne Bergnaum

12

0

2

# DIFF CALC THRY MATH 2400

Joanne Bergnaum
UGA
GPA 3.56

Staff

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COURSE
PROF.
Staff
TYPE
Class Notes
PAGES
2
WORDS
KARMA
25 ?

## Popular in Mathematics (M)

This 2 page Class Notes was uploaded by Joanne Bergnaum on Saturday September 12, 2015. The Class Notes belongs to MATH 2400 at University of Georgia taught by Staff in Fall. Since its upload, it has received 12 views. For similar materials see /class/202083/math-2400-university-of-georgia in Mathematics (M) at University of Georgia.

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Date Created: 09/12/15
Trigonometry New and Old 24 MATH 00H Fall 2006 De nitions 7 cos 9 7 sin 9 y r r y x r 0050 S600 I r x tang X ctn0 i x y Note that cos and sec are even functions whereas sin tan csc and cot are odd functions The basic relations are of course based on the Pythagorean Theorem sin2 0 cos2 0 1 tan2 0 1 2 sec2 0 1 cot2 0 csc2 0 Note also that sin 2 ox cos oz cos72 ox sin oz sin7 ox sin oz and cos7 ox cos a Area The area of a triangle is A bh If two sides a and b and the included angle 0 of a triangle are given then we take 17 as the base and the height is h a sin 0 so a 19 Addition Formulas We begin with T lsinot8 sinotcos cosotsin Consider the diagram at the right Here points D and E resp are obtained by dropping perpendiculars from B and A resp onto 0C It follows that AACE E ABCD and so Thus AECD BDCE from which we conclude that areaAACD areaABCE Now using this result we have1 sinot 8 2areaAOAB 2areaAOAC areaAOCB 2areaAOAD areaAOEB sinot cos8 COSOt sin 1This proof was inspired by a suggestion of Dr Mark Faucette at the University of West Georgia By substituting 8 for 8 in T and using the symmetries of the respective functions we obtain lsinot 8 sinotcos9 cosotsin8 Similarly substituting l at for at in T we obtain lcosot8 cosotcos9 qsinotsin8 As a consequence we have the double angle formulas 2 2 ot sin a2cos2a 11 2sin2a sin2a 2sinotcosot Law of Cosines In AABC let 0 be the length of the side opposite angle 0 Then 02a2b2 2abcos0 A The easiest proof is to assign cartesian coordinates to the points A B and C Let C 00 B a 0 and A b cos0 bsin0 Then we infer from the distance formula that lcos 205 cos 02 a bcos02 bsin02 a2 172 2abcos0 Law of Sines In AABC let the sides a b c be opposite the angles oz 8 y respectively Then sin oz sin 8 sin y a b c We ll establish only the rst equality here Drop a perpendicular from C to E as indicated in the diagram Then I I hbsinaasin gt wz a b as desired Basic Limits Recall that the area of the unit circle is by de nition 7 and so the area of a sector with central angle 0 is 7 0 From the gure at the right 7 we see rst that areaAOAC 5 areasector0AC and so sin0 5 0

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