Derivatives of trigonometric functions

Transcription

1 Derivatives of trigonometric functions 2 October 207 Introuction Toay we will ten iscuss te erivates of te si stanar trigonometric functions. Of tese, te most important are sine an cosine; te erivatives of all te oter stanar trigonometric follow reaily from tese. 2 Sine an cosine We ve seen ow to ifferentiate polynomials an eponential functions. Te oter main missing piece in te catalog of elementary functions tat can be easily ifferentiate are te trigonometric functions. Te two basic facts are te following. So tese two basic functions are closely linke to eac oter; te main confusing ting is to remember wic one obtains a negative sign wen it is ifferentiate. Te easiest way to get straigt on tis is just to tink about were te functions are increasing an ecreasing. Te grap of sine is initially increasing, so its erivative at 0 a better by positive; tus is must be an not. Similarly, te grap of starts at a local maimum, so its erivative but cange from positive to negative aroun 0; tis is te opposite of wat oes, so te erivative of must be an not. Observation. Bot sine an cosine ave te very special property tat tey are te negative of teir secon erivative, i.e. () an (). In pysical terms, eac curve is always accelerating back towars te ais at a rate given by its istance from te ais. Tis is te reason tat tese functions arise so muc in pysical problems: any system wit feeback tat pulls it back towars equilibrium (e.g. a weigt on a spring, or a swaying brige) is governe by some equations tat ultimately give rise to functions tat are built up from sine an cosine. Note. Te fact tat te erivatives of sine an cosine ave suc a nice form in terms of eac oter is te principle reason wy raians, rater tan egrees, are always use wen oing trigonometry (at least wen any tecniques from calculus are begin use). It is analogous to using te metric system in cemistry: just like te metric system makes unit conversions less error-prone, using raians makes taking erivatives less error-prone. Tis is eactly analogous to using e rater tan any oter eponential function. If you appen to ave seen te formula, in terms of comple numbers, e i + i, you will realize tat coosing e an coosing raians are really te eact same coice, if you take a sligtly broaer point of view.

2 2. Te erivation Te usual erivation of te erivatives of sine an cosine uses te following stanar trigonometric ientities (te wor ientity means a formula wic ol for all values of te input). I ve put everyting involving in blue to make it stan out. sin( + ) cos + cos( + ) cos Te basic facts tat allow us to compute te erivatives of sine an cosine are te following linear approimations to an : for for close to 0, Te first of tese was an ientity we iscusse in te lecture on linear approimation. Te secon follows because as a local maimum at 0, ence a orizontal tangent line. Applying tese linear approimations an te ientities above, we obtain te following fact: if is a very small number (very close to 0), ten: sin( + ) + cos( + ) From tese approimations, we see tat increases at a rate of (as increases), wile increases at a rate of. Te more formal version of wat I ave just sai is to first invoke te following two its (wic are bot compute by some analysis using te squeeze teorem, wic we will not escribe in etail). Bot are visually plausible if you raw te graps of sine an cosine; tey say tat te erivative of sine at 0 is, an tat te erivative of cosine at 0 is 0. 0 cos 0 Ten te erivative of can be compute as follows. 2

3 () sin( + ) cos + ( cos + ) cos Te erivative of can be formally commute in a totally analogous way. 2.2 A pysical interpretation () cos( + ) cos ( cos ) cos 0 Skip tis subsection if you on t particularly like pysics. But I fin te following picture to be te clearest eplanation for wy te erivatives of sine an cosine are wat tey are. In tis picture, you soul imagine te curve arc as a planet orbiting te origin ( unit away) in a perfect circle, traveling at spee eactly. Ten te velocity of tis orbiting planet will point in a irection tangent to te circle, an will ave magnitue. Ten you can etermine te an y coorinates of velocity by rawing te re triangle sown. It is congruent to te blue triangle, but rotate 90 egrees. Ten te erivatives of sine an cosine can be seen by immeiate visual inspection in tis picture. 3

4 3 Te si stanar trigonometric functions As you may ave seen in your precalculus, tere are si functions tat usually make up te stanar trigonometric functions. As follows. ypotenuse ajacent Function name Notation Definition sine opposite / ypotenuse tangent tan opposite / ajacent secant sec ypotenuse / ajacent opposite Function name Notation Definition cosine ajacent / ypotenuse cotangent cot ajacent / opposite cosecant csc ypotenuse / opposite Te nomenclature ere is a bit of a nigtmare, I m afrai. All tese terminology is couple unre years ol, an it one of tose vestigial organs tat we cannot seem to ecise from common usage. However, tis is te preominant nomenclature for tese functions so it is wort reviewing tem. Here s one feature tat makes all tis sligtly easier: tese si functions are arrange into four cofunction pairs: sine an cosine; tangent an cotangent; secant an cosecant. Eac function is relate to its co-function in a simple way: just swap ajacent an opposite werever you see tem 2. All si can be epresse in terms of sine an cosine, as follows. tan cot sec csc As a result, it is straigtforwar to ifferentiate all of tese functions by use of te quotient rule an te erivatives of sine an cosine. It is wort working tese computations yourself, as an eercise in te quotient rule. Te computations are sown below. I suggest tat you practice eriving tese formulas, even if you ultimately memorize tem. Tis is goo practice wit te quotient rule (wic will pay iviens in more comple computations), an also will elp you avoi overburening your brain wit arbitrary formulas. tan () () cos2 + sec 2 2 A more matematical way to say tis is: replace wit π/2 (so cos() sin( π 2 ), cot tan( π 2 csc sec( π 2 ). ), an 4

5 cot () () sin2 csc 2 0 () sec sec tan 0 () csc csc cot Note tat tere are generally many ways to write tese erivatives (for eample, te erivative of tan coul be written eiter or sec2 ). Te convention is generally to write functions witout enominators if possible; tis entails replacing wit sec were possible, for eample. By te way, ere s a elpful mnemonic for remembering some of tese erivatives, wic I calle informally in class te co-function rule 3. If you can remember te erivative of one function (e.g. if you remember tat tan sec2 ), ten you can obtain te erivative of it s co-function as follows: take te erivative of te original function, replace all functions tat appear by teir co-functions, an ten multiply by. For eample, if you know tat tan sec2, ten te cofunction rule tells you tat cot csc2. If you look over te si erivative we ave compute above, you will see tat tey all obey tis rule. 3 In fact, tis rule can be euce from te cain rule, I will leave you to tink about wy. 5