Inverse Trigonometric Functions - Domain, Range, and Graph of Cosec Inverse

• The sine function is defined as: $\sin(\theta) = \frac{1}{\csc^{-1}(\theta)}$
• The cosecant inverse function, denoted as $\csc^{-1}(\theta)$, is the inverse of the cosecant function $\csc(\theta)$
• In this lesson, we will discuss the domain, range, and graph of the cosecant inverse function

Domain of Cosec Inverse

• The domain of the cosecant inverse function is the set of all real numbers except:
  • Values where the cosecant function is zero (i.e., $\theta$ such that $\csc(\theta) = 0$)
  • Values that make the cosecant function undefined (i.e., $\theta$ such that $\sin(\theta) = 0$)
• We exclude these values to ensure that the inverse function is well-defined

Range of Cosec Inverse

• The range of the cosecant inverse function is the set of all real numbers except:
  • Values less than or equal to $-1$ (i.e., $y \leq -1$)
  • Values greater than or equal to $1$ (i.e., $y \geq 1$)
• We exclude these values to ensure that the inverse function is well-defined

Graph of Cosec Inverse

• The graph of the cosecant inverse function is symmetric with respect to the y-axis
• Asymptotes:
  • Vertical asymptotes occur at $x = 0$ and $x = \pi$
  • Horizontal asymptote is located at $y = 0$
• The curve is not defined for values outside the range of the cosecant inverse function

Example 1

Find the domain and range of the cosecant inverse function $\csc^{-1}(x)$.

• Domain: All real numbers except $x = 0$ and $x = \pi$
• Range: All real numbers except $y \leq -1$ and $y \geq 1$

Example 2

Graph the cosecant inverse function $\csc^{-1}(x)$.

Inverse Trigonometric Functions - Domain, Range, and Graph of Cosec Inverse

• Recap:
  • Domain: All real numbers except $x = 0$ and $x = \pi$
  • Range: All real numbers except $y \leq -1$ and $y \geq 1$
  • Graph: Symmetric with respect to the y-axis, vertical asymptotes at $x = 0$ and $x = \pi$, horizontal asymptote at $y = 0$
• Understanding the properties of the cosecant inverse function is essential for solving trigonometric equations and analyzing real-world problems

Inverse Trigonometric Functions - Domain, Range, and Graph of Sec Inverse

• The secant function is defined as: $\sec(\theta) = \frac{1}{\cos^{-1}(\theta)}$
• The secant inverse function, denoted as $\sec^{-1}(\theta)$, is the inverse of the secant function $\sec(\theta)$
• In this lesson, we will discuss the domain, range, and graph of the secant inverse function

Domain of Sec Inverse

• The domain of the secant inverse function is the set of all real numbers except:

  • Values where the secant function is zero (i.e., $\theta$ such that $\sec(\theta) = 0$)
  • Values that make the secant function undefined (i.e., $\theta$ such that $\cos(\theta) = 0$)

• We exclude these values to ensure that the inverse function is well-defined

Range of Sec Inverse

• The range of the secant inverse function is the set of all real numbers except:

  • Values less than or equal to $-1$ (i.e., $y \leq -1$)

  • Values greater than or equal to $1$ (i.e., $y \geq 1$)

• We exclude these values to ensure that the inverse function is well-defined

Graph of Sec Inverse

• The graph of the secant inverse function is symmetric with respect to the y-axis

• Asymptotes:

  • Vertical asymptotes occur at $x = -\frac{\pi}{2}$ and $x = \frac{\pi}{2}$

  • Horizontal asymptote is located at $y = 0$

• The curve is not defined for values outside the range of the secant inverse function

Example 1

Find the domain and range of the secant inverse function $\sec^{-1}(x)$.

• Domain: All real numbers except $x = -\frac{\pi}{2}$ and $x = \frac{\pi}{2}$
• Range: All real numbers except $y \leq -1$ and $y \geq 1$

Example 2

Graph the secant inverse function $\sec^{-1}(x)$.

Inverse Trigonometric Functions - Domain, Range, and Graph of Sec Inverse

• Recap:

  • Domain: All real numbers except $x = -\frac{\pi}{2}$ and $x = \frac{\pi}{2}$

  • Range: All real numbers except $y \leq -1$ and $y \geq 1$

  • Graph: Symmetric with respect to the y-axis, vertical asymptotes at $x = -\frac{\pi}{2}$ and $x = \frac{\pi}{2}$, horizontal asymptote at $y = 0$

• Understanding the properties of the secant inverse function is crucial for solving trigonometric equations and analyzing real-world problems

Inverse Trigonometric Functions - Domain, Range, and Graph of Cot Inverse

• The cotangent function is defined as: $\cot(\theta) = \frac{1}{\tan^{-1}(\theta)}$
• The cotangent inverse function, denoted as $\cot^{-1}(\theta)$, is the inverse of the cotangent function $\cot(\theta)$
• In this lesson, we will discuss the domain, range, and graph of the cotangent inverse function

Domain of Cot Inverse

• The domain of the cotangent inverse function is the set of all real numbers except:

  • Values where the cotangent function is undefined (i.e., $\theta$ such that $\tan(\theta) = 0$)

• We exclude these values to ensure that the inverse function is well-defined

Range of Cot Inverse

• The range of the cotangent inverse function is the set of all real numbers
• Unlike the other inverse trigonometric functions, the cotangent inverse function has no restrictions on its range
• The graph of the cotangent inverse function has no horizontal or vertical asymptotes
• The curve is defined for all real numbers

Inverse Trigonometric Functions - Domain, Range, and Graph of Secant Inverse

• The secant function is defined as:
  • $\sec(\theta) = \frac{1}{\cos^{-1}(\theta)}$
• The secant inverse function, denoted as $\sec^{-1}(\theta)$, is the inverse of the secant function $\sec(\theta)$
• In this lesson, we will discuss the domain, range, and graph of the secant inverse function

Domain of Secant Inverse

• The domain of the secant inverse function is the set of all real numbers except:
  • Values where the secant function is zero (i.e., $\theta$ such that $\sec(\theta) = 0$)
  • Values that make the secant function undefined (i.e., $\theta$ such that $\cos(\theta) = 0$)
• We exclude these values to ensure that the inverse function is well-defined

Range of Secant Inverse

• The range of the secant inverse function is the set of all real numbers except:
  • Values less than or equal to $-1$ (i.e., $y \leq -1$)
  • Values greater than or equal to $1$ (i.e., $y \geq 1$)
• We exclude these values to ensure that the inverse function is well-defined

Graph of Secant Inverse

• The graph of the secant inverse function is symmetric with respect to the y-axis
• Asymptotes:
  • Vertical asymptotes occur at $x = -\frac{\pi}{2}$ and $x = \frac{\pi}{2}$
  • Horizontal asymptote is located at $y = 0$
• The curve is not defined for values outside the range of the secant inverse function

Example 1

Find the domain and range of the secant inverse function $\sec^{-1}(x)$.

• Domain: All real numbers except $x = -\frac{\pi}{2}$ and $x = \frac{\pi}{2}$
• Range: All real numbers except $y \leq -1$ and $y \geq 1$

Example 2

Graph the secant inverse function $\sec^{-1}(x)$.

Inverse Trigonometric Functions - Domain, Range, and Graph of Cotangent Inverse

• The cotangent function is defined as:
  • $\cot(\theta) = \frac{1}{\tan^{-1}(\theta)}$
• The cotangent inverse function, denoted as $\cot^{-1}(\theta)$, is the inverse of the cotangent function $\cot(\theta)$
• In this lesson, we will discuss the domain, range, and graph of the cotangent inverse function

Domain of Cotangent Inverse

• The domain of the cotangent inverse function is the set of all real numbers except:
  • Values where the cotangent function is undefined (i.e., $\theta$ such that $\tan(\theta) = 0$)
• We exclude these values to ensure that the inverse function is well-defined

Range of Cotangent Inverse

• The range of the cotangent inverse function is the set of all real numbers
• Unlike the other inverse trigonometric functions, the cotangent inverse function has no restrictions on its range
• The graph of the cotangent inverse function has no horizontal or vertical asymptotes
• The curve is defined for all real numbers

Conclusion

• Inverse trigonometric functions play an important role in solving trigonometric equations and analyzing real-world problems
• Understanding the properties of these functions, including their domains, ranges, and graphs, helps us make meaningful interpretations and calculations
• By studying inverse trigonometric functions, we expand our mathematical toolkit and enhance our problem-solving skills