68 Chapter 3Transcendental functions
Special values
The values for the angles on the boundaries of the four quadrants, and for a few other
angles, are listed in Table 3.2.
Table 3.2 Some special values
θ 0° 90° 180° 2 70° 360° 30° 60° 45°
θ 2 rad 0 π 22 π 3 π 222 ππ 26 π 23 π 24
sin 1 θ 01 0− 10122
cos 1 θ 10 −10 1 122
tan 1 θ 0 ∞ 0 −∞ 01
EXAMPLE 3.5Demonstrate thatsin 1 π 221 = 11 andcos 1 π 221 = 1 0.
By Figure 3.8,
sin 1 θ 1 = 1 a 2 c,cos 1 θ 1 = 1 b 2 c
As the angle θincreases to 90°, the magnitude of side bapproaches
zero whilst that of side aaproach the value of c. Therefore
sin 1 θ 1 → 1 c 2 c 1 = 1 1andcos 1 θ 1 → 102 c 1 = 1 0asθ 1 → 1 π 22
0 Exercises 7
EXAMPLE 3.6Verify the values of the trigonometric functions for θ 1 = 1 π 26 in
Table 3.2.
Draw an equilateral triangle with sides of length 2 and bisect the triangle as in
Figure 3.9. Then, by Pythagoras, , andsin 1 π 261 = 1122 ,cos 1 π 261 =1,
tan 1 π 261 = 1
0 Exercise 8
13
h=−=21 3 32
223
1332 12
12
32..............................................................................................................................................................................................................................................................
...
...
..
....
..
...
...
...
...
...
...
..
....
..
...
...
...
...
...
...
..
....
..
...
...
...
...
...
...
...
...
..
...
...
...
...
...
...
...
...
..
...
...
...
...
...
...
...
...
..
....
..
...
...
...
...
...
...
...
...
..
...
...
.........................................................................................................................................................................................................................................................................................
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
................................................................................60◦30◦h
2
1
2
1
Figure 3.9
Figure 3.8
................................................................................................................................................................................................................................................................
...
...
....
...
...
...
...
...
...
...
...
....
...
...
...
...
...
...
...
...
....
...
...
...
...
...
....
...
...
...
...
...
...
...
...
....
...
...
...
...
...
...
...
....
...
...
...
...
...
....
...
...
...
...
...
...
...
..................................................................θ
a
c
b