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7.1 Sines and Cosines and their Deriaties

A hi poin e claim ha he deiaie of he ine fncion i anohe elaed fncion called he coine fncion. Thi

ma eem meio o o ince e hae no defined eihe of hee fncion, b e ill eenall define hem.

The cosine function, whose value at argument x is generally written as cos(x) is a short way of saying the sineof the complementary angle to x. The complemena angle o i he diffeence beeen a igh angle and he angle

.

We ill ala o meae angle in adian, hogh o of habi, old fa like melf ofen lape ino decibing

hem b degee.

Imagine e hae an angle a ome poin P and da a cicle C aond P ha ha adi 1. Then the size of the

angle in radians is the length of the arc between the end-lines of on the circle C.

I i an impoan bi of folkloe ha he oal diance aond a cicle of ni adi i . Th he ie of an angle i

he popoion of he cicle ha i epeen, mliplied b he faco .

We can heefoe ee ha a aigh line angle epeen half a cicle o i ha

angle hile a igh angle, hich i half of a aigh line, ha angle . The complemena angle o i he angle

.

So he deiaie of he ine (ien all a in) obe

b he chain le, e ge

Thee fac abo he deiaie of he ine and coine ae almo a imple a hoe fo he eponen, and he ae no

difficl o e in pacice. Fo moe deail abo igonomeic fncion, click here.

Amed ih hee la o fac e can e he biion le and o peio le o diffeeniae an fncion e

cae o conc fom he ideni, he eponen, and he ine b aihmeic opeaion and biion.

Are we done?

We ae almo done. Pacice ing he mliple occence le and he chain le a bi and o can become an epe

diffeeniao. B e ill hae o noice ho o diffeeniae inee fncion.

Exercises:

7.1 Find the derivative for each of the following functions:

a. (sin x) * exp(2x)

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b. * c 2

c. (c ) * i

7.2 Chec ae ih heae.

7.3 The deiaie f he deiaie f a fci f i caed he ecd deiaie f f. Fid he ecd

deiaie f c , ad a f i 2.

The ecd deiaie f f i a deed b f ''() .

7.4 Wha fci ca hi f ha be he eai f() + f ''() = 0?

Wh d a ih he ie fci eeia ad idei? ha haeed he cie? O

he he igeic fci? Ad ha ae he eid hig caca ie ch ad ih?

We don' bohe ih eaing he coine epaael ince e can define i fom he ine b biion:

. The ohe igonomeic fncion can be defined fom hee o. The fncion ih he h on

he end ae called hpebolic ine and coine. The ae eail epeed in em of he eponenial fncion:

The ohe fncion ha appea on good calclao and ae aailable on peadhee ae eail conced fom hoe

menioned o fa o fom hei inee.

Ad ha ae iee?

We ill ee ha no, and alo ho o find hei deiaie.

U Pei Ne