Jim Lambers

Math 1B

Fall Quarter 2004-05

Leture 10 Notes

These notes orrespond to Setion 5.4 in the text.

Trigonometri Funtions

In Leture 8, we dened the irular funtions using the values of the wrapping funtion. All of

these funtions dened various relations b etween a real number

x

and the point (

a; b

) on the unit

irle that is reahed by traveling a distane of

x

in the ounterlokwise diretion around the irle.

In Leture 9, we introdued angles, whih desrib ed relationships between rays that shared a

ommon vertex. To measure angles, we dened the degree and the radian, whih quan

tied the

extent to whih an angle's initial side needed to rotate in order to meet its terminal side.

Let

P

be a point on the initial side of an angle that lies one unit from the angle's vertex. As

the angle is formed by the rotation of the initial side toward the terminal side, the point

P

travels

along an ar of a irle of radius one, and it travels a distane that is equal to the radian measure

of the angle. It is natural to ask what the oordinates of the point

P

are, one it has reahed the

terminal side. It turns out that the irular funtions an help us to answer this question.

Denition of the Trigonometri Funtions

Suppose that the vertex of an angle lies at the origin, and that the initial side of the angle oinides

with the p ositive

x

-axis. Then, the p oint

P

desribed in the previous paragraph, that lies one unit

from the vertex of the initial side, is the point (1

;

0), whih is a point on the unit irle.

As the initial side rotates toward the angle's terminal side, the point

P

travels along the unit

irle for a distane equal to the radian measure of the angle, as previously disussed. In this ase,

however, beause the p oint

P

began its journey at (1

;

0), we atually know the oordinates of

P

one it has rotated to the terminal side: it is equal to

W

(

x

), the value of the wrapping funtion at

x

, where

x

is the radian measure of the angle.

Beause the irular funtions are dened in terms of the wrapping funtion, and we have shown

how the wrapping funtion an be assoiated with an angle, we an dene the irular funtions

in suh a way that their domains belong to the set of all angles, rather than the set of all real

numbers. In this ontext, the irular funtions are alled

trigonometri funtions

.

W

e now preisely dene the six trigonometri funtions that assoiate angles with points on

the unit irle. First, we must emphasize that we an only make this assoiation if we require that

angles have their vertex at (0

;

0) and have initial sides that oinide with the positive

x

-axis. We

therefore introdue a term that desribes suh angles, for onveniene.

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