Tan(90°)

Definition

The expression tan(90°) represents the tangent of a 90-degree angle, which is undefined within the context of trigonometric functions. This undefined nature arises because, at 90 degrees, the corresponding point on the unit circle is (0, 1), leading to a situation where the tangent ratio, defined as the ratio of the opposite side to the adjacent side, has an adjacent side of zero. Consequently, tangent becomes undefined as you cannot divide by zero.

5 Must Know Facts For Your Next Test

1. The tangent function can be represented as tan(θ) = sin(θ)/cos(θ), and at 90°, cos(90°) = 0, making tan(90°) undefined.
2. On the unit circle, at 90°, the coordinates are (0, 1), which means there is no horizontal distance to create a tangent ratio.
3. Tangent has vertical asymptotes at odd multiples of 90 degrees, including tan(90°), which signifies that it approaches infinity but never reaches a defined value.
4. The behavior of tan(θ) around θ = 90° shows that as angles approach 90°, the values of tangent increase significantly, indicating instability in this function.
5. In practical applications, encountering tan(90°) indicates situations where slope or angles are critical, such as in physics or engineering, but these cases must be handled with care due to undefined behavior.

Review Questions

• What happens to the tangent function as it approaches 90 degrees and why is tan(90°) considered undefined?
  • As angles approach 90 degrees, the values of tangent increase dramatically and approach infinity because the cosine value becomes zero, causing division by zero in the tangent formula. Since division by zero is undefined in mathematics, this is why tan(90°) itself is categorized as undefined. This behavior illustrates a key feature of the tangent function and its vertical asymptote at this angle.
• How do the properties of the unit circle help explain why tan(90°) is undefined?
  • The unit circle defines trigonometric functions based on coordinates associated with angles. At 90 degrees, the coordinates are (0, 1), leading to a situation where there is no horizontal distance (adjacent side), resulting in an undefined tangent ratio. The unit circle provides clarity on how each angle corresponds to specific x and y coordinates, and for tan(90°), since it involves division by zero, it is classified as undefined.
• Evaluate how understanding tan(90°) impacts our interpretation of trigonometric functions and their applications in real-world scenarios.
  • Understanding that tan(90°) is undefined highlights critical concepts within trigonometric functions and their limitations. In real-world applications like physics or engineering, recognizing when an angle leads to undefined values is essential for accurate calculations. This comprehension not only shapes our mathematical knowledge but also aids in developing strategies for handling situations where such values occur, ensuring we can make informed decisions based on mathematical principles.

Related terms

Tangent: A trigonometric function that relates the angle to the ratio of the length of the opposite side to the length of the adjacent side in a right triangle.

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

Undefined: A term used in mathematics to indicate that a particular value does not have a meaning or cannot be calculated.