Range of sinθ and cosθ

Range of cosθ and sinθ

Let P(x, y) be a point on the unit circle, so $OP = 1$, and let $\angle AOB = \theta$ (with B the terminal ray through P). Since P lies on the unit circle,

Because $y^2 \ge 0$, this forces $x^2 \le 1$; similarly $y^2 \le 1$. Taking square roots,

and since $x=\cos\theta$, $y=\sin\theta$ on the unit circle, this is exactly

for every real θ. In other words, both $\sin\theta$ and $\cos\theta$ take values only in the closed interval $[-1,1]$ — they never exceed 1 in magnitude, and (as the graphs later confirm) every value in that interval is actually attained.

Solved Example — signs of sin300°, cos400°, cot(−206°)

To find the sign of a trigonometric function of an angle outside $[0°, 360°)$, first reduce it to a coterminal angle in that range (coterminal angles share the same terminal ray, hence the same trigonometric values), then read the sign off the quadrant it falls in.

i) sin 300°. Since $270° < 300° < 360°$, the angle 300° already lies in the fourth quadrant. In the fourth quadrant sinθ is negative (only cosθ is positive there). So sin 300° is negative. …