chap1 - Chapter 1 Introductory material Last revised 9...

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Chapter 1 Introductory material Last revised 9 October 2008 This chapter gives a quick review of some of those parts of the prerequisite courses (Calculus I and II and Geometry I) which we will actually use, adding some extra material. Those parts which are revision will be without examples. 1.1 Trigonometric functions 1.1.1 Values (See Thomas 1.6) We can quickly obtain the value of a trigonometric function for any argument in terms of values for x [ 0 , 1 2 π ] by remembering a few things. First we have the table 0 30 = π 6 radians 45 = π 4 rad. 60 = π 3 rad. 90 = π 2 rad. cos 1 3 2 1 2 1 2 0 sin 0 1 2 1 2 3 2 1 To get the sign for other values we can use the mnemonic table Radians Degrees sin cos tan Positive functions ( 0 , 1 2 π ) ( 0 , 90 ) + + + All ( 1 2 π , π ) ( 90 , 180 ) + Sin ( π , 3 2 π ) ( 180 , 270 ) + Tan ( 3 2 π , 2 π ) ( 270 , 360 ) + Cos sometimes called the ‘Add Sugar To Coffee’ rule – or use Thomas’ variant “All Students Take Calculus”. (Note: to be entirely accurate we should have special rows in this table for the values 1 2 π etc because at those points one or more of the functions will be zero or unbounded.) 1
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Then we remember what happens when we replace x by x , x + π / 2 or x + π : cos ( x ) = cos x , sin ( x ) = sin x , cos ( x + π 2 ) = sin x , sin ( x + π 2 ) = cos x , (1.1) cos ( x + π ) = cos x , sin ( x + π )= sin x . These are very easy to derive from e ix = cos x + i sin x , remembering that e i π / 2 = i , e i π = 1. Using them in combination we can get cos ( π x ) = cos x , sin ( π x ) = sin x . and so on. More generally cos ( x + ( 2 m + 1 ) π 2 ) = ( 1 ) ( m + 1 ) sin x , sin ( x + ( 2 m + 1 ) π 2 ) = ( 1 ) m cos x , (1.2) cos ( x + n π ) = ( 1 ) n cos x , sin ( x + n π ) = ( 1 ) n sin x . (1.3) where m and n are integers. These identities enable us to relate the value we want to a value in the first quadrant (i.e. the range [ 0 , 1 2 π ] ). Note in particular cos ( n π ) = ( 1 ) n , sin (( 2 m + 1 ) π / 2 )= ( 1 ) m . (1.4) which will turn up later on. 1.1.2 Identities for the trigonometric functions The most important formulae to remember are sin 2 A + cos 2 A = 1 (1.5) cos ( A + B ) = cos A cos B sin A sin B (1.6) sin ( A + B ) = sin A cos B + cos A sin B . (1.7) If you have trouble remembering which of the last two is which, and which has the minus in it, try substituting some special values such as A = 0 or B = 1 2 π and checking the result. For example, taking A = 0 in the last equation gives sin B = sin B whereas if you had tried sin ( A + B ) = sin A cos B cos A sin B you would get sin B = sin B . From these and the earlier results we get cos ( A B ) = cos A cos B + sin A sin B sin ( A B ) = sin A cos B cos A sin B . and we can use these to get cos A cos B = 1 2 ( cos ( A + B )+ cos ( A B )) (1.8) sin A sin B = 1 2 ( cos ( A B ) cos ( A + B )) (1.9) sin A cos B = 1 2 ( sin ( A + B )+ sin ( A B )) , (1.10) which we will find very useful in doing integrations like integraltext cos ( nx ) cos ( mx ) d x .
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