Calculus: Analytic Geometry and Calculus, with Vectors

14 Analytic geometry in two dimensions

usual assumptions about them. We can hear many different and even contradic-
tory tales about the world, but we can always be cheered by the fact that our
assumptions are universally considered to be interesting enough and useful enough
to be worthy of study. Absorbing these ideas may not keep us young and fair, but
we need the ideas to be debonair.

1.2 Slopes and equations of lines When we study trigonometry, we

learn about the plane rectangular coordinate system shown in Figures

Figure 1.21 Figure 1.22

1.21 and 1.22 and we become familiar with the formulas

s

sing=y, tang=y, sec 0 =

(1.23) r x x

r

cos0=x, cot0=x, csc 0 =

r y y

which define the six basic trigonometric functions.t In each figure, the

horizontal axis is the x axis, the vertical axis is the y axis, and the inter-

section 0 of the two axes is the origin of the coordinate system. Since the

F(-2,3) TT

J(2,3)

A(3,2)

E(0 2)

II I

I(-3,0)

C(4,0)

I InIivI I 1 i

G(3,-1)

B(-4,-2) D(0,-2) H(4,

.

x

matter will be of great importance to

us, we review the standard procedure

for plotting (or locating) points whose

coordinates are given. To plot the

point 4(3,2), the point .4 whose coor-

dinates are the positive numbers 3

and 2, we start at the origin and go 3

units to the right (in the direction of

the positive x axis) and then go 2

units up (in the direction of the posi-

Figure 1.24 tive y axis) to reach the point 4 of

Figure 1.24. To plot the point B

whose coordinates are the negative numbers -4 and -2,we start at the

origin and go 4 units to the left (in the direction of the negativex axis)

and then go 2 units down (in the direction of the negativey axis) to reach

B. Everyone should examine Figure 1.24 to see that the other pointsare

correctly plotted. The signs of the coordinates tell us whichways we go,

f Our rigorous presentation of angles and trigonometric functions willcome in Chapter 8.

Meanwhile we shall very often review anduse facts about angles and trigonometric func-

tions that are learned in trigonometry.

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