James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

A26
appendix D Trigonometry
hypotenuse
opposite
The Trigonometric Functions
For an acute angle the six trigonometric functions are defined as ratios of lengths of
sides of a right triangle as follows (see Figure 6).
¨
FIGURE 6
adjacent
4 sin − opp
hyp
cos − adj
hyp
tan − opp
adj
csc − hyp
opp
sec − hyp
adj
cot − adj
opp
P(x, y)
r
y
¨
This definition doesn’t apply to obtuse or negative angles, so for a general angle
in stan dard position we let Psx, yd be any point on the terminal side of and we let r be
the dis tance | OP | as in Figure 7. Then we define
O
x
5 sin − y r
csc − r y
FIGURE 7
cos − x r
sec − r x
tan − y x
cot − x y
If we put r − 1 in Definition 5 and
draw a unit circle with center the origin
and label as in Figure 8, then the
coordinates of P are scos , sin d.
P(cos ¨, sin ¨)
FIGURE 8
y
1
O
¨
1
x
Since division by 0 is not defined, tan and sec are undefined when x − 0 and csc
and cot are undefined when y − 0. Notice that the definitions in (4) and (5) are consistent
when is an acute angle.
If is a number, the convention is that sin means the sine of the angle whose radian
measure is . For example, the expression sin 3 implies that we are dealing with an angle
of 3 rad. When finding a calculator approximation to this number, we must remember
to set our calculator in radian mode, and then we obtain
sin 3 < 0.14112
If we want to know the sine of the angle 38 we would write sin 38 and, with our calculator
in degree mode, we find that
sin 38 < 0.05234
The exact trigonometric ratios for certain angles can be read from the triangles in
Fig ure 9. For instance,
sin 4 − 1
s2
sin 6 − 1 2
sin 3 − s3
2
œ„ 2
π
4
1
FIGURE 9
π
4 2 π
1
3 1
π
6
œ„3
cos 4 − 1
s2
cos 6 − s3
2
tan 4 − 1 tan 6 − 1
s3
cos 3 − 1 2
tan 3 − s3
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