[-1, 1]

Definition

The interval [-1, 1] refers to the set of real numbers between and including -1 and 1. This interval is particularly important in the context of inverse trigonometric functions, as it represents the domain and range of these functions.

5 Must Know Facts For Your Next Test

1. The interval [-1, 1] is the domain and range of the inverse trigonometric functions, such as $\arcsin(x)$, $\arccos(x)$, and $\arctan(x)$.
2. The inverse trigonometric functions are used to find the angle when given the value of a trigonometric function, such as finding the angle whose sine is 0.5.
3. The interval [-1, 1] represents the set of values for which the trigonometric functions sine, cosine, and tangent are defined.
4. The inverse trigonometric functions are restricted to the interval [-1, 1] to ensure that they are one-to-one functions, which means that each output value corresponds to a unique input value.
5. The interval [-1, 1] is important in the context of inverse trigonometric functions because it guarantees that the functions are well-defined and have a unique solution for any value in the interval.

Review Questions

• Explain the significance of the interval [-1, 1] in the context of inverse trigonometric functions.
  • The interval [-1, 1] is the domain and range of the inverse trigonometric functions, such as $\arcsin(x)$, $\arccos(x)$, and $\arctan(x)$. This interval is important because the trigonometric functions sine, cosine, and tangent are only defined for values within this range. By restricting the domain and range of the inverse functions to [-1, 1], we ensure that they are one-to-one functions, meaning that each output value corresponds to a unique input value. This allows the inverse functions to be well-defined and have a unique solution for any value in the interval.
• Describe how the interval [-1, 1] is related to the definition and properties of the trigonometric functions.
  • The interval [-1, 1] is directly related to the definition and properties of the trigonometric functions. The sine, cosine, and tangent functions are defined for angles in a right triangle, and the values of these functions are restricted to the range [-1, 1]. For example, the sine function takes on values between -1 and 1, inclusive. This means that the inverse sine function, $\arcsin(x)$, is only defined for values of $x$ within the interval [-1, 1]. The same is true for the inverse cosine function, $\arccos(x)$, and the inverse tangent function, $\arctan(x)$. The interval [-1, 1] is a fundamental property of the trigonometric functions and their inverses.
• Analyze the importance of the interval [-1, 1] in the context of solving problems involving inverse trigonometric functions.
  • The interval [-1, 1] is crucial when solving problems involving inverse trigonometric functions. Since the inverse functions are only defined for values within this interval, it is essential to ensure that the input values fall within the appropriate range. If the input value is outside of [-1, 1], the inverse function will not be well-defined, and the problem cannot be solved correctly. Additionally, the properties of the inverse functions, such as their domains and ranges, are directly tied to the interval [-1, 1]. Understanding the significance of this interval and how it relates to the inverse trigonometric functions is crucial for successfully solving a wide range of problems in mathematics and physics that involve these functions.

Related terms

Domain:

The set of input values for which a function is defined.

Range:

The set of output values that a function can produce.

Trigonometric Functions: The functions sine, cosine, tangent, and their inverses, which describe the relationships between the sides and angles of a right triangle.