$[-1, 1]$

Definition

$[-1, 1]$ is a closed interval on the real number line that includes all real numbers between and including -1 and 1. This interval is commonly used in the context of the unit circle, which is a fundamental concept in trigonometry.

5 Must Know Facts For Your Next Test

1. The interval $[-1, 1]$ represents all the possible $x$-coordinates of points on the unit circle.
2. The $y$-coordinates of points on the unit circle are also bounded by the interval $[-1, 1]$.
3. The endpoints of the interval, -1 and 1, correspond to the points on the unit circle where the angle is 180 degrees and 0 degrees, respectively.
4. The values within the interval $[-1, 1]$ represent the range of possible values for the trigonometric functions sine and cosine.
5. The unit circle is often used to visualize and understand the behavior of trigonometric functions, and the interval $[-1, 1]$ is a key feature of this representation.

Review Questions

• Explain how the interval $[-1, 1]$ is related to the unit circle and its use in trigonometry.
  • The interval $[-1, 1]$ is directly related to the unit circle, as it represents the range of possible $x$-coordinates and $y$-coordinates of points on the unit circle. This interval is crucial in trigonometry because the trigonometric functions sine and cosine are defined using the $x$-coordinates and $y$-coordinates of points on the unit circle, respectively. The values within the $[-1, 1]$ interval correspond to the range of possible values for these trigonometric functions, making the unit circle and the $[-1, 1]$ interval essential tools for understanding and visualizing trigonometric concepts.
• Describe how the endpoints of the interval $[-1, 1]$ relate to specific points on the unit circle and their corresponding angles.
  • The endpoints of the interval $[-1, 1]$ correspond to specific points on the unit circle. The point where the $x$-coordinate is -1 represents an angle of 180 degrees, while the point where the $x$-coordinate is 1 represents an angle of 0 degrees. These endpoints are significant because they mark the boundaries of the range of possible $x$-coordinates and $y$-coordinates on the unit circle, which in turn define the range of values for the trigonometric functions. Understanding the relationship between the $[-1, 1]$ interval and the corresponding angles on the unit circle is crucial for applying trigonometric concepts in various mathematical and scientific contexts.
• Analyze how the interval $[-1, 1]$ is used to determine the values of the trigonometric functions sine and cosine, and explain the significance of this relationship.
  • The interval $[-1, 1]$ is directly linked to the values of the trigonometric functions sine and cosine, as these functions are defined using the $y$-coordinates and $x$-coordinates of points on the unit circle, respectively. The range of values for sine and cosine is bounded by the interval $[-1, 1]$, meaning that the sine and cosine functions will always return values within this range. This relationship is significant because it allows for the visualization and understanding of trigonometric functions, as well as the ability to use the unit circle to solve various problems involving angles and their corresponding trigonometric values. The $[-1, 1]$ interval is a fundamental concept that underpins the application of trigonometry in fields such as physics, engineering, and mathematics.

Related terms

Interval: A set of real numbers that includes all values between a lower and upper bound, which can be either inclusive or exclusive.

Unit Circle: A circle with a radius of 1 unit, centered at the origin (0, 0) on the coordinate plane, used to define trigonometric functions.

Trigonometric Functions: Functions that describe the relationships between the sides and angles of a right triangle, such as sine, cosine, and tangent.