Page 182 - Mathematics_Form_Two

Trigonometry

Figures 8.13 and 8.14 illustrate how
Solution
positive and negative angles can be located
(a) 165° is in the second quadrant, then in the four quadrants. The corresponding
=
cos 165° − cos(180° − 165°) positive and negative angles whose
= − cos 15° trigonometric ratios are the same can
(b) 317° is in the fourth quadrant, then easily be found.
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Mathematics for Secondary Schools (c) 95° is in the second quadrant, then anticlockwise
=
sin 317° − sin (360° − 317°)
=

− sin 43°
direction
=
tan 95° − tan (180° − 95°)
=

− tan 85°
(d) 258° is in the third quadrant, then
tan 258° = tan (258° − 180°)
= tan 78°

Positive and negative angles
Angles may be positive or negative Figure 8�13: Positive angles in a unit circle
depending on the direction in which
the angle is measured. Figure 8.2
gives the clockwise and anticlockwise
measurements of an angle.

y
P

i
clockwise
O − θ x direction

Figure 8�14: Negative angles in a unit circle
Q
If θ is positive, the negative angle
Figure 8�12: Measurements of angles in a clockwise
and anticlockwise directions corresponding to θ is ( 360 θ− + ). If θ is
negative, the positive angle corresponding
In Figure 8.12, it can be deduced that, angles to θ is (360 θ+ ).
measured in the clockwise direction from
the positive x-axis are negative. Angles Note: sin(−θ) =−sinθ, cos(−θ) = cosθ,
measured in the anticlockwise direction
from the positive x-axis are positive. and tan(−θ) =−tanθ.

176
Student's Book Form Two

11/10/2024 20:14:01
MATHEMATIC F2 v5.indd 176
MATHEMATIC F2 v5.indd 176 11/10/2024 20:14:01