Trigonometric Functions c 2002 Donald Kreider and Dwight Lahr

Transcription

1 Trigonomeric Funcions c 2002 Donald Kreider and Dwigh Lahr Modeling wih Trigonomeric Funcions: You firs me he rigonomeric funcions in algebra and rigonomery in high school. In a ypical rigonomery course he funcions sin x, cos x, and he oher relaed funcions an x, sec x, ec., are defined as raios of sides in righ riangles. The focus is on measuring he sides and angles of riangles, hence he erm rigonomeric funcions. Much of your aenion was direced o applicaions in geomery growing ou of his connecion wih righ riangles as well as o ideniies ha express relaionships beween he several rigonomeric funcions. In calculus he focus changes. The rigonomeric funcions are defined in erms of arclengh on a uni circle, and he emphasis is on he periodic behavior of he rigonomeric funcions. I is heir periodiciy ha leads o heir mos imporan applicaions in science modeling phenomena ha repea as a funcion of ime. Simple harmonic moion (sinusoidal moion), ligh and sound waves, elecriciy, graviaional waves in he universe, oscillaions of he pendulum of a clock, oscillaions of aomic crysals on which our mos accurae ime keeping is based all hese are periodic phenomena ha are modeled mahemaically by he rigonomeric funcions. The essenial characerisics of any periodic moion are he ampliude (how big are he oscillaions?) and period (how long is a single wave?). Someimes he phase (how much has he wave been ranslaed o he righ or delayed in ime?) is also imporan, as when one is comparing wo relaed wave forms such as volage and curren in he sandard way of generaing elecriciy he curren lags he volage by 90 degrees (he phase angle is π/2. Their graphs migh look as follows: volage curren Here he ampliude of he volage is.4, he ampliude of he curren is, boh have period 2π, and he curren lags he volage by a phase angle of π/2. The period ells us how long a single wave is, and, especially when he independen variable is ime, his is someimes described insead in erms of frequency (how many oscillaions occur in a uni disance or uni of ime). Definiion : The Trigonomeric Funcions: In calculus we define he rigonomeric funcions in erms of arc lengh on a uni circle. Choose a poin on he circle (see he plo below), a a disance from he posiive x-axis measured counerclockwise along he circle. Then he rigonomeric funcions cos and sin are defined o be he coordinaes of he poin.

2 2 sin (co s, sin ) cos The arclengh in he plo is a measure of he cenral angle ha subends he arc. I is called he radian measure of he angle. An angle of 360 subends he enire circle whose lengh is 2π, hus 360 = 2π radians, and = π/80 radians. Also 80 = π radians, 90 = π/2 radians, and 60 = π/3 radians We will consisenly use radian measure of angles in calculus. Reference o he figure also ells us ha our new definiion of he rigonomeric funcions is consisen wih he usual definiions as raios of sides of a righ riangle. In he small righ riangle in he figure wih angle (radians), he hypoenuse of he riangle is, he adjacen side has lengh cos, and he opposie side has lengh sin. Thus, for example, he raio of he adjacen side o he hypoenuse is cos, as i should be. The main hing o remember is ha we are measuring angles in radians insead of degrees, so cos and sin are funcions of he real variable. There is hus no longer any reason o resric our aenion o angles less han 360. We may measure any posiive disance along he uni circle ha we wish, and since he poin on he circle reurns o is saring poin when = 2π, we noe ha he values of sin and cos repea when we wrap around he circle, i.e. when increases by a muliple of 2π. This gives us he fundamenal periodic behavior of sin and cos, he propery ha makes hem so useful in modeling periodic phenomena. We can also allow o ake on negaive values. These simply correspond o disances measured clockwise along he circle, beginning a he poin (, 0). The sin and cos funcions hus have domain < <. Apple: Definiions of sin(x)and cos(x) Try i! Theorem : The rigonomeric funcions sin and cos are defined for all real values of, and are periodic wih period 2π. I.e. hey saisfy sin ( + n 2π) = sin for any real and any ineger n. Many of he familiar properies of he rigonomeric funcions follow immediaely from heir definiion. Since (sin, cos ) is a poin on he uni circle i is clear ha sin 2 + cos 2 = for all values of. (Noe: we follow he usual convenion of wriing sin 2 insead of he more cumbersome (sin ) 2 ; however, when doing calculaions, we do ype he laer so ha a compuer can undersand i.) Moreover i is also clear ha cos ( ) = cos and sin ( ) = sin (see he figure below), hus cos is an even funcion and sin is an odd funcion. These symmeries and he periodiciy, along wih special values such as cos 0 =, cos π/2 = 0, cos π =, sin 0 = 0, sin π/2 =, ec., enable us o skech heir graphs:

3 3 y = cos (co s, sin ) - y = sin ' (co s(- ), sin(- )) Oher special values of sin and cos can be read direcly from he geomery of he uni circle. For example he angle π/4 (or 45 ) deermines an isosceles righ riangle wih hypoenuse ; hus he legs of his riangle boh have lengh / 2. I follows ha cos π/4 = sin π/4 = / 2. In making such compuaions i is ofen more convenien o use a reference riangle ha is similar o he small one in he uni circle bu wih more convenien dimensions. In his example i is easier o refer o a riangle wih legs of lengh and hypoenuse of lengh 2 and o read off he values of he rigonomeric funcions using heir radiional definiions as raios of sides of he riangle. Reference riangles for several special angles are shown below. From hese can you read off, for example, he values of sin π/6, cos π/6, sin π/3, and cos π/3? π/ π/4 π/3 The examples above deal wih angles in he firs quadran (0 π/2). Angles in oher quadrans can be handled using, again, he geomery of he uni circle. An angle in he second quadran, for example, can be refleced in he y-axis and compared wih he angle π in he firs quadran. Examining he figure below i is clear ha cos (π ) = cos and sin (π ) = sin. In paricular, herefore, cos (5π/6) =- cos(π/6), and he value can hen be read off from one of he reference riangles shown above. Angles in he hird quadran can be refleced in he origin and compared wih an angle in he firs quadran. And angles in he fourh quadran can be refleced in he x-axis. By considering only angles in he firs quadran, herefore, and using he reference riangle mehod for compuing sin and cos of special angles, we can compue he values of hese rigonomeric funcions for angles of any size. (co s( π-), sin( π-)) (co s( π+), sin( π+)) - (co s, sin ) (co s(- ), sin(- )) h a π/2 - b We should menion, finally, he ideniies cos (π/2 ) = sin and sin (π/2 ) = cos. The angles and π/2 are complemenary. They share a single righ riangle. Do you see ha he raio a h is cos as well as sin (π/2 )? As a general rule, i is no necessary o remember hundreds of unrelaed facs and ideniies for he rigonomeric funcions. A lile undersanding of he geomery involved goes a long way.

4 4 Definiion: Oher rigonomeric funcions: For he record we define he oher rigonomeric funcions ha are ofen used. They are an = sin cos co = an = cos sin sec = csc = cos sin Noice ha all of hese addiional rigonomeric funcions are defined in erms of sin and cos. And, in fac, he sine funcion also can be defined in erms of cos since, for angles in he firs quadran sin = cos 2. In a sense, hen, mos of he rigonomeric funcions are superfluous only one of hem is needed, and all he ohers can be defined in erms of ha one. Bu his would obscure he rich se of ideniies relaing he differen rigonomeric funcions. I is his algebraic richness ha can be exploied in solving many problems. We do no inend o pause here for an exensive survey of such ideniies, nor do we recommend memorizing endless liss of ideniies and properies. Raher, we have menioned a few basic ideniies ha follow more or less direcly from he uni circle definiion, and we prefer o derive more ideniies only as he need for hem arises. Example : Suppose θ is a hird quadran angle and cos θ = /4. Find an θ. Refer o he figure below. We firs observe ha he sign of an θ is posiive since boh sin and cos are negaive in he hird quadran. Then, reflecing he poin in he origin, we may use he more convenien reference riangle shown in he figure o compue he value of an θ. Thus an θ = + an = 5. The values of he oher rigonomeric funcions can also be compued from his reference riangle, aking ino accoun heir signs in he hird quadran. So sin θ = 5/4, sec θ = 4, ec.. -/4 θ 4 5 Example 2: Skech he graph of an x. Since an x = sin x cos x, we noice ha he x-inerceps of he graph occur where sin x = 0, i.e. a he poins x = nπ where n is an ineger. Moreover an x is undefined whenever cos x = 0. Thus he graph has verical asympoes x = π/2 + nπ, n an ineger. Example 3: Skech he graph of f(x) = + sin 2x. We can visualize his graph as he graph of sin x compressed by he facor 2 (all horizonal disances muliplied by /2) and shifed up uni. Moreover since sin x has period 2π, he funcion f(x) has period π. We show he graphs of boh funcions below.

5 5 y= + sin( 2x ) y=sin x Apple: Trigonomeric Ideniies Try i! Exercises: roblems Check wha you have learned! Videos: Tuorial Soluions See problems worked ou!