Lesson 6 of 13 · 7 min
Signs in the four quadrants
NCERT §3.3.1
As the wheel turns, Meera's cabin is sometimes east of the hub and sometimes west, sometimes above and sometimes below. Those two facts alone decide the sign of all six functions.
The lesson in notes
In short
If P(a, b) is reached by turning through x, turning through −x reaches Q(a, −b), the mirror image of P in the x-axis. So cos (−x) = cos x and sin (−x) = −sin x, and hence tan (−x) = −tan x.
Every point of the unit circle has both coordinates between −1 and 1, so −1 ≤ sin x ≤ 1 and −1 ≤ cos x ≤ 1 for every x.
The signs follow the quadrant of P. In the first quadrant (0 < x < π/2) both coordinates are positive; in the second (π/2 < x < π) a < 0 < b; in the third (π < x < 3π/2) both are negative; in the fourth (3π/2 < x < 2π) b < 0 < a.
Hence sin x > 0 for 0 < x < π and sin x < 0 for π < x < 2π, while cos x > 0 in the first and fourth quadrants and cos x < 0 for π/2 < x < 3π/2.
Positive functions by quadrant: all six in the first, only sin and cosec in the second, only tan and cot in the third, only cos and sec in the fourth. A reciprocal always shares the sign of its partner.
To find all six values from one, fix the size with an identity and the sign with the quadrant. If sin x = 8/17 with x in the second quadrant, cos² x = 1 − 64/289 = 225/289, and cos x < 0 there, so cos x = −15/17, tan x = −8/15, cosec x = 17/8, sec x = −17/15 and cot x = −15/8.
Second example: if cos x = 7/25 with x in the fourth quadrant, sin² x = 1 − 49/625 = 576/625 and sin x < 0 there, so sin x = −24/25 and tan x = −24/7.