Trigonometric functions: characteristics of sine, cosine and tangent

The sine function, $sin(x)$

  1. Domain: $\mathbb{R}$
  2. Image: $[-1,1]$
  3. Period: $2\pi$ rad
  4. Continuity: It is continuous on $\mathbb{R }$
  5. Increasing on: $\ldots \cup \Big(\displaystyle -\frac{\pi}{2}, \frac{\pi}{2}\Big) \cup \Big(\displaystyle \frac{3\pi}{2}, \frac{5\pi}{2}\Big)\cup \ldots$
  6. Decreasing on: $\ldots \cup \Big(\displaystyle \frac{\pi}{2}, \frac{3\pi}{2}\Big) \cup \Big(\displaystyle \frac{5\pi}{2}, \frac{7\pi}{2}\Big)\cup \ldots$
  7. Maxima at: $\Big\{ \displaystyle \frac{\pi}{2}+2\pi\cdot k$, $k \in \mathbb{Z}\Big\}$
  8. Minima at: $\Big\{ \displaystyle \frac{3\pi}{2}+2\pi\cdot k$, $k \in \mathbb{Z}\Big\}$
  9. Parity: Odd, $\sin x=-\sin (-x)$
  10. Points of intersection with the axis Ox: $x=k\cdot \pi$, $k \in \mathbb{Z}$

The cosine function, $cos(x)$

  1. Domain: $\mathbb{R}$
  2. Image: $[-1,1]$
  3. Period: $2\pi$ rad
  4. Continuity: It is continuous on $\mathbb{R }$
  5. Increasing on: $\ldots \cup (-\pi,0) \cup (\pi,2\pi) \cup \ldots$
  6. Decreasing on: $\ldots \cup (0,\pi) \cup (2\pi,3\pi) \cup \ldots$
  7. Maxima at: $\Big\{ 2\pi\cdot k$, $k \in \mathbb{Z}\Big\}$
  8. Minima at: $\Big\{ \pi\cdot (2k+1)$, $k \in \mathbb{Z}\Big\}$
  9. Parity: Pair $\cos x = \cos (-x)$
  10. Points of intersection with the axis Ox: $x=\displaystyle \frac{\pi}{2}+k \cdot \pi$, $k \in \mathbb{Z}$

The tangent function, $tan(x)$

  1. Domain: $\mathbb{R}-\Big\{ (2k+1) \cdot \displaystyle \frac{\pi}{2}, k \in \mathbb{Z}\Big\}=\mathbb{R}- \Big\{ \ldots, \displaystyle -\frac{\pi}{2},\frac{\pi}{2}, \frac{3\pi}{2}, \ldots \Big\}$
  2. Image: $\mathbb{R}$
  3. Period: $\pi$ rad
  4. Continuity: It is continuous on $\mathbb{R}-\Big\{\displaystyle \frac{\pi}{2}+k\pi, k \in \mathbb{Z} \Big\}$
  5. Increasing on: $\mathbb{R}$
  6. Maxima: No maxima
  7. Minima: No minima
  8. Parity: Odd $\tan x = - \tan (-x)$
  9. Points of intersection with the axis Ox: $x=k\cdot \pi, k \in \mathbb{Z}$

Practice exercises