Plane trigonometry with practical applications, by Leonard E. Dickson.

152 TRIGONOMETRY [Art. 107
Y
91200 ~ _ sine curve
150~/ '\30~
xtl — 0 ___ Pto o 0_ 0\ 210o240"270'000o330~3600 X
180~ '_360 00 300 600 90~10 10~ 10
210 1 o_ A
27 0 -
FIG. 85. SINE AND COSINE CURVES (ART. 107)
00
300 If angle A is in the first or fourth quadrant (Figs.
81, 84), sec A =r/x is represented by the radius vector
60-7 OT to the point in the terminal side of A whose ab90o / scissa x is OX = 1, If A is in the second or third
quadrant (Figs. 82, 83), sec A is equal to the negative
120, of the ratio of the length of OT to OX = 1; since T
1.5 1, is on the prolongation of the terminal side of A, it is
_. customary to regard OT as negative, while lines in
180 the direction OP along the terminal side of A are
l f D m ~ positive. Hence the secant of an angle in its trigonometric position is equal, in magnitude and sign, to the
\ _ ) directed line OT from the origin to the point T in which
\ 7 f the terminal side of the angle, prolonged if necessary,
intersects the line tangent to the unit circle at the point X
3000 > on the initial side of the angle, prpvided OT is counted
positive or negative according as T is on the terminal side
3300 or on its prolongation.
3600 Similarly, the cosecant of an angle in its trigonometric
position is equal, in magnitude and sign, to the directed
line OT' from the origin to the point T' in which the
Z terminal side, prolonged if necessary (and then OT' is
counted negative), intersects the line tangent to the unit circle at the point Y on
the positive y-axis.
107. The sine and cosine curves. These curves may be drawn
rapidly by means of the line representations just discussed. Draw
a circle whose radius is any convenient length representing unity.
Divide the circumference of the circle into 12 equal parts and
label the points of division 0~, 30~, 60~,..., 330~ (Fig. 85). Through