key term - Quadrantal Angle

Definition

A quadrantal angle is a special type of angle that is a multiple of 90 degrees, or a quarter of a full revolution around the unit circle. These angles are particularly important in the context of trigonometric functions, as they exhibit unique properties and behaviors.

5 Must Know Facts For Your Next Test

1. Quadrantal angles are multiples of 90 degrees, or $\pi/2$ radians, and include 0°, 90°, 180°, and 270°.
2. At quadrantal angles, the trigonometric functions sine and cosine exhibit unique values, with sine taking on the values 0, 1, 0, and -1, and cosine taking on the values 1, 0, -1, and 0, respectively.
3. Quadrantal angles are particularly useful in simplifying trigonometric expressions and evaluating trigonometric functions, as the values of sine and cosine are easily identifiable.
4. The unit circle is often divided into four quadrants, with each quadrant corresponding to a range of 90 degrees or $\pi/2$ radians.
5. Quadrantal angles are important in the study of periodic functions, as they represent points where the function repeats or exhibits symmetry.

Review Questions

• Explain the significance of quadrantal angles in the context of the unit circle and trigonometric functions.
  • Quadrantal angles, which are multiples of 90 degrees or $\pi/2$ radians, are particularly important in the study of the unit circle and trigonometric functions. At these angles, the sine and cosine functions exhibit unique and easily identifiable values, making them useful for simplifying trigonometric expressions and evaluating trigonometric functions. Additionally, the unit circle is often divided into four quadrants, each corresponding to a range of 90 degrees, and quadrantal angles represent key points where the trigonometric functions exhibit symmetry or repetition.
• Describe how the values of sine and cosine change at quadrantal angles on the unit circle.
  • At quadrantal angles on the unit circle, the values of the sine and cosine functions exhibit a repeating pattern. Specifically, at 0 degrees (or 0 radians), the sine value is 0 and the cosine value is 1. At 90 degrees (or $\pi/2$ radians), the sine value is 1 and the cosine value is 0. At 180 degrees (or $\pi$ radians), the sine value is 0 and the cosine value is -1. Finally, at 270 degrees (or $3\pi/2$ radians), the sine value is -1 and the cosine value is 0. This pattern of sine and cosine values at quadrantal angles is crucial for understanding and simplifying trigonometric expressions.
• Analyze the relationship between quadrantal angles, periodic functions, and the symmetry of trigonometric functions.
  • Quadrantal angles are closely linked to the periodic nature of trigonometric functions and the symmetry exhibited by these functions. Since the trigonometric functions, such as sine and cosine, repeat their values every 360 degrees or $2\pi$ radians, the quadrantal angles of 0°, 90°, 180°, and 270° represent key points where the functions exhibit symmetry. For example, the sine function is an odd function, meaning it is symmetric about the origin, and this symmetry is evident at the quadrantal angles. Similarly, the cosine function is an even function, symmetric about the y-axis, and this symmetry is also clearly visible at the quadrantal angles. Understanding the relationship between quadrantal angles and the periodic and symmetric properties of trigonometric functions is crucial for analyzing and working with these functions.

Related terms

Unit Circle:

A circular representation of the trigonometric functions, where the radius of the circle is 1 unit and the angle is measured in radians or degrees.

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

Radian:

A unit of angle measurement based on the circumference of a circle, where one radian is the angle subtended by an arc equal in length to the radius of the circle.