3.2 Trigonometric functions 67
The trigonometric functions are defined for all points on the circle as
(3.10)
In the first quadrant, I, with angles 01 < 1 θ 1 < 1 π 22 , the values of xand yare both positive
and
sin 1 θ 1 > 1 0cos 1 θ 1 > 1 0tan 1 θ 1 > 10
In the second quadrant however, withπ 221 < 1 θ 1 < 1 π, xis negative and yis positive. The
signs of the trigonometric functions in all four quadrants are given in Table 3.1.
Table 3.1 Signs of the trigonometric functions
Quadrant I II III IV
Angles 01 < 1 θ 1 < 1 π 22 π 221 < 1 θ 1 < 1 ππ 1 < 1 θ 1 < 13 π 223 π 221 < 1 θ 1 < 12 π
01 < 1 θ 1 < 1 90° 90° 1 < 1 θ 1 < 1 180° 180° 1 < 1 θ 1 < 1 270° 270° 1 < 1 θ 1 < 1 360°
sin 1 θ ++ − −
cos 1 θ +− − +
tan 1 θ +− + −
EXAMPLE 3.4By Table 3.2, sin 1 π 231 = 1 , cos 1 π 231 = 112 2,and tan 1 π 231 = .What
are the values of the sine, cosine, and tangent of 2 π 2 3, 4π 23 ,and 5 π 23?
The angle 2 π 23 is in the second quadrant, 4 π 23 in the third, 5 π 23 in the fourth.
Therefore, by Figure 3.7,
sin 12 π 231 = 1 + cos 12 π 231 = 1 − 12 2tan 12 π 231 = 1 −
sin 14 π 231 = 1 − cos 14 π 231 = 1 − 12 2tan 14 π 231 = 1 +
sin 15 π 231 = 1 − cos 15 π 231 = 1 + 12 2tan 15 π 231 = 1 −
0 Exercises 6
32 3
32 3
32 3
32 3
sinθθθ===cos tan
y
r
x
r
y
x
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- •
2 π/ 3 π/ 3
4 π/ 35 π/ 3
0
π
− −
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Figure 3.7