# Domain and range of trigonometric functions

Understanding the domain and range of trigonometric functions is essential for working with these functions in various mathematical contexts. The domain represents the set of all possible input values, while the range is the set of all possible output values. Here's a breakdown for some common trigonometric functions:

1. Sine Function ($\mathrm{sin}$):

• Domain:
• The domain of $\mathrm{sin}\left(\theta \right)$ is all real numbers.
• $\theta$ can take any real value, as the sine function is defined for any angle.
• Range:
• The range of $\mathrm{sin}\left(\theta \right)$ is $\left[-1,1\right]$.
• The sine function produces values between -1 and 1, inclusive.

2. Cosine Function ($\mathrm{cos}$):

• Domain:
• The domain of $\mathrm{cos}\left(\theta \right)$ is all real numbers.
• $\theta$ can take any real value, as the cosine function is defined for any angle.
• Range:
• The range of $\mathrm{cos}\left(\theta \right)$ is $\left[-1,1\right]$.
• The cosine function produces values between -1 and 1, inclusive.

3. Tangent Function ($\mathrm{tan}$):

• Domain:
• The domain of $\mathrm{tan}\left(\theta \right)$ is all real numbers except for values where $\theta$ is odd multiples of $\frac{\pi }{2}$(e.g., $\frac{\pi }{2},\frac{3\pi }{2},\frac{5\pi }{2},\dots$).
• The tangent function is undefined at these points due to division by zero.
• Range:
• The range of $\mathrm{tan}\left(\theta \right)$ is all real numbers.
• The tangent function can take any real value.

4. Cotangent Function ($\mathrm{cot}$):

• Domain:
• The domain of $\mathrm{cot}\left(\theta \right)$ is all real numbers except for values where $\theta$ is multiples of $\pi$ (e.g., $0,\pi ,2\pi ,\dots$).
• The cotangent function is undefined at these points due to division by zero.
• Range:
• The range of $\mathrm{cot}\left(\theta \right)$ is all real numbers.
• The cotangent function can take any real value.

5. Secant Function ($\mathrm{sec}$):

• Domain:
• The domain of $\mathrm{sec}\left(\theta \right)$ is all real numbers except for values where $\theta$ is odd multiples of $\frac{\pi }{2}$ (e.g., $\frac{\pi }{2},\frac{3\pi }{2},\frac{5\pi }{2},\dots$).
• The secant function is undefined at these points due to division by zero.
• Range:
• The range of $\mathrm{sec}\left(\theta \right)$ is $\left(-\mathrm{\infty },-1\right]\cup \left[1,+\mathrm{\infty }\right)$.
• The secant function produces values less than or equal to -1 and greater than or equal to 1.

6. Cosecant Function ($\mathrm{csc}$):

• Domain:
• The domain of $\mathrm{csc}\left(\theta \right)$ is all real numbers except for values where $\theta$ is multiples of $\pi$ (e.g., $0,\pi ,2\pi ,\dots$).
• The cosecant function is undefined at these points due to division by zero.
• Range:
• The range of $\mathrm{csc}\left(\theta \right)$ is $\left(-\mathrm{\infty },-1\right]\cup \left[1,+\mathrm{\infty }\right)$.
• The cosecant function produces values less than or equal to -1 and greater than or equal to 1.

7. Periodicity:

• All trigonometric functions have periodic behavior, repeating their values after certain intervals (periods). For sine and cosine, the period is $2\pi$, and for tangent, cotangent, secant, and cosecant, the period is $\pi$.