Sec(90°)

Definition

The term sec(90°) refers to the secant of an angle measuring 90 degrees. In trigonometry, secant is defined as the reciprocal of the cosine function. At 90 degrees, the cosine value is zero, which means the secant function approaches infinity, indicating that sec(90°) is undefined. This concept highlights important properties of trigonometric functions, especially how they behave at specific angles on the unit circle.

5 Must Know Facts For Your Next Test

1. Secant is denoted as sec(x), where x is the angle in question, and it is calculated as sec(x) = 1/cos(x).
2. At 90 degrees, cos(90°) = 0, making sec(90°) = 1/0, which results in an undefined value.
3. The behavior of sec(90°) illustrates that certain angles lead to vertical asymptotes in the graph of the secant function.
4. Understanding sec(90°) helps highlight how the trigonometric functions interact with one another and can lead to undefined values.
5. Sec(90°) being undefined is significant when solving equations involving secant and understanding limits in calculus.

Review Questions

• What happens to the value of sec(90°) and how does it relate to the cosine function?
  • The value of sec(90°) is undefined because it equals 1 divided by cos(90°), and since cos(90°) is 0, this results in a division by zero situation. This relationship shows that understanding the properties of cosine is essential when analyzing secant values. The undefined nature at this angle emphasizes critical points where trigonometric functions may not have valid outputs.
• How does the concept of undefined values in trigonometric functions like sec(90°) inform our understanding of their graphs?
  • The undefined nature of sec(90°) indicates that there are vertical asymptotes in its graph at this angle. Graphing secant reveals that it has regions where it cannot be evaluated due to its dependence on cosine. These vertical asymptotes occur at odd multiples of 90 degrees where cosine equals zero, helping us visualize how trigonometric functions behave around critical angles.
• Analyze how the behavior of sec(90°) relates to broader concepts in trigonometry and calculus, particularly around limits.
  • The behavior of sec(90°) being undefined connects directly to broader trigonometric concepts by illustrating how certain inputs lead to asymptotic behavior. In calculus, examining limits approaching 90 degrees for sec(x) shows that as x approaches 90 from either side, sec(x) approaches positive or negative infinity. This behavior is crucial when discussing continuity and differentiability in trigonometric functions, highlighting where these functions can be continuous or exhibit vertical asymptotes.

Related terms

Cosine: A fundamental trigonometric function defined as the ratio of the adjacent side to the hypotenuse in a right triangle or as the x-coordinate of a point on the unit circle.

Reciprocal: A mathematical term referring to a number which, when multiplied by a given number, yields one; for example, the reciprocal of x is 1/x.

Unit Circle: A circle with a radius of one centered at the origin of a coordinate plane, which is used to define trigonometric functions for all angles.