Domain and Range of Trigonometric functions

Domain and Range of Trigonometric functions

Here sinθ, cosθ, tanθ (and their reciprocals) are treated as functions of a real variable θ, measured in radians. Two coterminal angles α and θ (with $0°\le\alpha\le360°$) always give $\sin\theta=\sin\alpha$ and $\cos\theta=\cos\alpha$, confirming that sine and cosine are defined for every real θ.

i) $y=\sin\theta$. Every real θ corresponds to some point on the unit circle, so sine is defined for all $\theta\in\mathbb{R}$: domain = ℝ. Since $\sin^2\theta+\cos^2\theta=1$ forces $-1\le\sin\theta\le1$ (§2.1.3), and every value in between is attained as θ sweeps around the circle: range = [−1, 1].

ii) $y=\cos\theta$. By the same reasoning: domain = ℝ, range = [−1, 1].

iii) $y=\tan\theta$. Since $\tan\theta = y/x$ with $x=\cos\theta$, tanθ fails to exist exactly where $\cos\theta=0$, i.e. at $\theta = \pm\pi/2, \pm3\pi/2, \pm5\pi/2, \dots$ — in general at $\theta=(2n+1)\pi/2,\ n\in\mathbb{Z}$. So domain = ℝ except $\theta=(2n+1)\pi/2$. As θ approaches $\pi/2$ from below, $\tan\theta\to+\infty$; approaching from above, $\tan\theta\to-\infty$. Since tanθ is a ratio of the two otherwise-unrestricted coordinates x and y, it can equal any real number: range = ℝ.

iv) $y=\text{cosec}\,\theta = 1/\sin\theta$. Undefined wherever $\sin\theta=0$, i.e. at $\theta=0,\pm\pi,\pm2\pi,\dots$, in general $\theta=n\pi,\ n\in\mathbb{Z}$. So domain = ℝ except $\theta=n\pi$. Since $-1\le\sin\theta\le1$, its reciprocal satisfies $\text{cosec}\,\theta\ge1$ or $\text{cosec}\,\theta\le-1$: range = $\{y\in\mathbb{R}:|y|\ge1\}$ = ℝ − (−1, 1). …