3.2 Trigonometric functions 69
Negative angles
Each point on the circle can be reached by either anti-clockwise rotation or by
clockwise rotation. An angle is defined to have positive value for an anti-clockwise
rotation and negative for a clockwise rotation.
The point P in Figure 3.10, corresponding to the negative angle −θ, can be reached
by anti-clockwise rotation through angle(2π 1 − 1 θ), and the two angles have the same
trigonometric values:
sin(−θ) 1 = 1 sin(2π 1 − 1 θ) 1 = 1 −sin 1 θ
cos(−θ) 1 = 1 cos(2π 1 − 1 θ) 1 = 1 +cos 1 θ (3.11)
tan(−θ) 1 = 1 tan(2π 1 − 1 θ) 1 = 1 −tan 1 θ
Further angles
The range of allowed values of the angle can be extended further by allowing one or
more complete rotations around the centre. Each complete rotation adds or subtracts
2 π, and the anglesθ 1 ± 12 πn, for all values of the integern 1 = 1 0, 1, 2, 3 =, have the same
trigonometric values:
sin(θ 1 ± 12 πn) 1 = 1 sin 1 θ,cos(θ 1 ± 12 πn) 1 = 1 cos 1 θ (3.12)
In addition, the tangent repeats every half rotation,
tan(θ 1 ± 1 πn) 1 = 1 tan 1 θ (3.13)
We see that, whereas every angle corresponds to a point on the circle, each point cor-
responds to an infinite number of angles. The graphs of sine, cosine, and tangent are
shown in Figure 3.11.
33The graph of the sine function was first drawn in 1635 by Gilles Personne de Roberval (1602–1675).
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Figure 3.10