Introduction
Trigonometric functions are an important part of mathematics, especially in geometry, algebra, and calculus. To understand these functions clearly, we need to know their domain (the input values allowed) and range (the output values obtained). This guide explains the domain and range of all six basic trigonometric functions with examples and graphs.
1. Domain and Range of Sine Function (sin x)
Domain: \(\displaystyle x \in \mathbb{R}\)
Range: \(\displaystyle [-1, \, 1]\)
Example:
\(\displaystyle \sin(0) = 0\)
\(\displaystyle \sin(90^{\circ}) = 1\)
\(\displaystyle \sin(270^{\circ}) = -1\)
Graph: The sine curve oscillates between \(\displaystyle -1\) and \(\displaystyle 1\).
2. Domain and Range of Cosine Function (cos x)
Domain: \(\displaystyle x \in \mathbb{R}\)
Range: \(\displaystyle [-1, \, 1]\)
Example:
\(\displaystyle \cos(0) = 1\)
\(\displaystyle \cos(180^{\circ}) = -1\)
Graph: The cosine curve also oscillates between \(\displaystyle -1\) and \(\displaystyle 1\).
3. Domain and Range of Tangent Function (tan x)
Domain: All real numbers except \(\displaystyle x = \frac{(2n+1)\pi}{2}, \; n \in \mathbb{Z} \quad (\text{where } \cos x = 0)\)
Range: \(\displaystyle (-\infty, \, \infty)\)
Example:
\(\displaystyle \tan(45^{\circ}) = 1\)
\(\displaystyle \tan(90^{\circ}) \; \to \; \text{undefined}\)
Graph: Tangent has vertical asymptotes where \(\displaystyle \cos x = 0\).
4. Domain and Range of Cosecant Function (csc x)
Domain: All real numbers except \(\displaystyle x = n\pi, \; n \in \mathbb{Z} \quad (\text{where } \sin x = 0)\)
Range: \(\displaystyle (-\infty, -1] \cup [1, \infty)\)
Example:
\(\displaystyle \csc(30^{\circ}) = 2\)
\(\displaystyle \csc(90^{\circ}) = 1\)
Graph: Cosecant has gaps where \(\displaystyle \sin x = 0\).
5. Domain and Range of Secant Function (sec x)
Domain: All real numbers except \(\displaystyle x = \frac{(2n+1)\pi}{2}, \; n \in \mathbb{Z} \quad (\text{where } \cos x = 0)\)
Range: \(\displaystyle (-\infty, -1] \cup [1, \infty)\)
Example:
\(\displaystyle \sec(0) = 1\)
\(\displaystyle \sec(180^{\circ}) = -1\)
Graph: Secant function has branches extending beyond \(\displaystyle \pm 1\).
6. Domain and Range of Cotangent Function (cot x)
Domain: All real numbers except \(\displaystyle x = n\pi, \; n \in \mathbb{Z} \quad (\text{where } \sin x = 0)\)
Range: \(\displaystyle (-\infty, \infty)\)
Example:
\(\displaystyle \cot(45^{\circ}) = 1\)
\(\displaystyle \cot(90^{\circ}) = 0\)
Graph: Cotangent has vertical asymptotes where \(\displaystyle \sin x = 0\).
Summary Table
| Function | Domain | Range |
|---|---|---|
| \(\displaystyle \sin x\) | \(\displaystyle \mathbb{R}\) | \(\displaystyle [-1,1]\) |
| \(\displaystyle \cos x\) | \(\displaystyle \mathbb{R}\) | \(\displaystyle [-1,1]\) |
| \(\displaystyle \tan x\) | \(\displaystyle \mathbb{R} - \left\{ \frac{(2n+1)\pi}{2} \right\}\) | \(\displaystyle (-\infty, \infty)\) |
| \(\displaystyle \csc x\) | \(\displaystyle \mathbb{R} - \{n\pi\}\) | \(\displaystyle (-\infty, -1] \cup [1, \infty)\) |
| \(\displaystyle \sec x\) | \(\displaystyle \mathbb{R} - \left\{ \frac{(2n+1)\pi}{2} \right\}\) | \(\displaystyle (-\infty, -1] \cup [1, \infty)\) |
| \(\displaystyle \cot x\) | \(\displaystyle \mathbb{R} - \{n\pi\}\) | \(\displaystyle (-\infty, \infty)\) |
Conclusion
The domain and range of trigonometric functions are key to solving trigonometry problems effectively. Sine and cosine remain between -1 and 1, while tangent and cotangent cover all real values except their undefined points. Understanding these properties helps students easily solve equations, graph functions, and apply trigonometry in real-life contexts.
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FAQs
Ans: Because sine and cosine are based on the unit circle, where the maximum and minimum values are 1 and -1.
Ans: \(\displaystyle \text{Because } \tan x = \frac{\sin x}{\cos x}, \;\text{and } \cos x = 0 \;\text{at these points. Division by zero is undefined.}\)
Ans: Sine and cosine are bounded between -1 and 1, while secant and cosecant extend beyond ±1.
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