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6
CHAPTER 1. TRIGONOMETRY
(g) If P is in the first quadrant, sin θ > 0, cos θ > 0, tan θ > 0.
If P is in the second quadrant, sin θ > 0, cos θ < 0, tan θ < 0.
If P is in the third quadrant then sin θ < 0, cos θ < 0, tan θ > 0.
If P is in the fourth quadrant then sin θ < 0, cos θ > 0, tan θ < 0.
It follows that if 0 < θ < π then sin θ > 0 while if π < θ < 2π then sin θ < 0.
Also if −π/2 < θ < π/2, then cos θ > 0, while if π/2 < θ < 3π/2 then cos θ < 0.
Finally, if 0 < θ < π/2 or π < θ < 3π/2 then tan θ > 0, while if π/2 < θ < π or
3π/2 < θ < 2π then tan θ < 0. All this information is summarized in Figure 3.2.2.
II
I
sin θ > 0
cos θ < 0
tan θ < 0
sin θ > 0
cos θ > 0
tan θ > 0
sin θ < 0
cos θ < 0
tan θ > 0
sin θ < 0
cos θ > 0
tan θ < 0
III
IV
Figure 3.2.2
(h) If θ = π/4, then x = y = sin θ = cos θ.
Since
sin2 (π/4) + cos2 (π/4) = 1
by (1.4), then
so
2 sin2 (π/4) = 1,
1
sin π/4 = cos π/4 = √ , and tan π/4 = 1.
2
............. P
√ 6 1 ....
..
1/ 2
..
? π/4 ...
A
O¾ √1/ 2
Figure 3.2.3
(i) If θ =
π
(= 60◦ ), then ∠OP B = 30◦ = π/6 (see Figure 3.2.4). Hence
3
OB =
1
1
OP = .
2
2