upgrade

➗calculus ii review

key term - $ ext{sin}^{-1}$

Citation:

Definition

$ ext{sin}^{-1}$ is the inverse sine function, which is used to find the angle whose sine is a given value. It represents the angle in the standard position of a unit circle that has a given sine value. This term is particularly relevant in the context of integrals resulting in inverse trigonometric functions, as the inverse sine function can arise in the solutions to certain types of integrals.

5 Must Know Facts For Your Next Test

  1. The inverse sine function, $ ext{sin}^{-1}$, is used to find the angle whose sine is a given value, rather than finding the sine of a given angle.
  2. The domain of the inverse sine function is the interval $[-1, 1]$, as the sine function is only defined for values within this range.
  3. The range of the inverse sine function is the interval $ ext{left}[- rac{ ext{pi}}{2}, rac{ ext{pi}}{2} ext{right}]$, as these are the possible angles whose sine values lie between -1 and 1.
  4. The inverse sine function is often used in integrals involving the sine function, as it can be used to simplify the integration process.
  5. The inverse sine function is denoted as $ ext{sin}^{-1}(x)$ or $ ext{arcsin}(x)$, and it represents the angle in standard position whose sine is $x$.

Review Questions

  • Explain the relationship between the sine function and the inverse sine function.
    • The sine function and the inverse sine function are inverse functions of each other. The sine function takes an angle as input and returns the corresponding sine value, while the inverse sine function, $ ext{sin}^{-1}$, takes a sine value as input and returns the angle whose sine is that value. This means that if $y = ext{sin}(x)$, then $x = ext{sin}^{-1}(y)$, and vice versa. The inverse sine function allows you to find the angle given the sine value, which is particularly useful in the context of integrals involving the sine function.
  • Describe the domain and range of the inverse sine function, $ ext{sin}^{-1}$, and explain their significance.
    • The domain of the inverse sine function, $ ext{sin}^{-1}$, is the interval $[-1, 1]$, as the sine function is only defined for values within this range. The range of the inverse sine function is the interval $ ext{left}[- rac{ ext{pi}}{2}, rac{ ext{pi}}{2} ext{right}]$, which represents the possible angles whose sine values lie between -1 and 1. These restrictions on the domain and range are important because they ensure that the inverse sine function is a well-defined and single-valued function, which is necessary for its use in various mathematical contexts, including integrals involving the sine function.
  • Explain how the inverse sine function, $ ext{sin}^{-1}$, can be used to simplify the integration of expressions involving the sine function.
    • The inverse sine function, $ ext{sin}^{-1}$, can be used to simplify the integration of expressions involving the sine function. This is because the integral of $ ext{sin}(x)$ can be expressed in terms of the inverse sine function, $ ext{sin}^{-1}(x)$. Specifically, the integral of $ ext{sin}(x)$ can be written as $- ext{sin}^{-1}(x) + C$, where $C$ is the constant of integration. This relationship allows the inverse sine function to be used to evaluate integrals involving the sine function, which is particularly useful in the context of integrals resulting in inverse trigonometric functions, as discussed in Section 1.7.