30-60-90 and
45-45-90 Degree Triangles
The trigonometric function values for some common angles should
be committed to memory. Consider the 30-60-90 and 45-45-90 degree triangles
shown in Figures 20.54
A
and
20.54
B
.
By looking at Figure 20.54
A
and Figure 20.54
B
we can readily find the values of the six trigonometric
functions for 30, 60, and 45 degrees and generate the entries for Table 20.5
Trigonometric Function Values for 0 and
90 Degrees
The trigonometric function values for 0 and 90 degrees are more
abstract. To visualize the trigonometric functions for these angles we draw
imaginary right triangles in which one side has a length of 0. The triangle for
0 degrees is shown in Figure 20.55.
The triangle shown in Figure 20.56 shows the imaginary triangle
used to find the values for the trigonometric functions of 90 degrees.
The Unit Circle
A circle with radius 1 drawn on Cartesian coordinates can be a
useful way to visualize values for trigonometric functions of angles greater
than 90 degrees. This circle is called a
unit circle
and is shown in Figure 20.57 with a right triangle in
the first quadrant.
Quadrant I
Using any point on the unit circle, a right triangle can be
constructed by dropping a perpendicular from the point on the circle to the
x
-axis. The hypotenuse of any
right triangle drawn this way will always be a radius of the circle and therefore
equal to 1. Accordingly, the trigonometric functions of any resulting angle
will simply be:
Notice that in Quadrant I all of the
x
and
y
values are positive so all of the
trigonometric functions yield positive results. We can also readily confirm the
values for the trigonometric functions for 0 and 90 degrees:
Quadrant II
Angles between 90 and 180 degrees fall in Quadrant II, where all
of the
x
values are negative and all the
y
values are positive, as shown in
Figure 20.58.
Inspection of Figure 20.58 reveals that for angles between 90
and 180 degrees:
For the special case of 180 degrees, Figure 20.58 shows that
where opposite side is 0, while adjacent and hypotenuse are 1.
Quadrant III
Angles between 180 and 270 degrees fall in Quadrant III where
all of the
x
and
y
values are negative as shown in Figure 20.59.
Inspection of Figure 20.59 reveals that for angles between 180
and 270 degrees:
For the special case of 270 degrees, Figure 20.59 shows that:
Quadrant IV
Angles between 270 and 360 degrees fall in Quadrant IV where all
of the x values
are positive and all the y values are negative as shown in
Figure 20.60.
Inspection of Figure 20.60 reveals that for angles between 270
and 360 degrees:
EXERCISES
20.51
Complete
the table by stating for each quadrant whether the indicated
trigonometric function is positive or negative.
20.52
Use a scientific calculator to find the value of each
trigonometric function.
Copyright © 2006
Industrial Press Inc.