In Exercises 9–16, evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined.tan 𝜋2

Verified step by step guidance

1. Identify the angle given in the problem, which is \( \frac{\pi}{2} \). This is a quadrantal angle, meaning it lies on one of the axes in the unit circle.
2. Recall that the tangent function, \( \tan(\theta) \), is defined as \( \frac{\sin(\theta)}{\cos(\theta)} \).
3. Evaluate \( \sin\left(\frac{\pi}{2}\right) \) and \( \cos\left(\frac{\pi}{2}\right) \). On the unit circle, \( \sin\left(\frac{\pi}{2}\right) = 1 \) and \( \cos\left(\frac{\pi}{2}\right) = 0 \).
4. Substitute these values into the tangent function: \( \tan\left(\frac{\pi}{2}\right) = \frac{1}{0} \).
5. Since division by zero is undefined, conclude that \( \tan\left(\frac{\pi}{2}\right) \) is undefined.

Key Concepts

Quadrantal Angles

Quadrantal angles are angles that are multiples of 90 degrees (or π/2 radians) and correspond to the axes in the unit circle. These angles include 0, π/2, π, 3π/2, and 2π. At these angles, the sine and cosine functions take on specific values, which are essential for evaluating trigonometric functions.

Tangent Function

The tangent function is defined as the ratio of the sine and cosine functions: tan(θ) = sin(θ) / cos(θ). It represents the slope of the line formed by the angle in the unit circle. At certain angles, particularly quadrantal angles, the tangent function can be undefined if the cosine value is zero, leading to division by zero.

Undefined Expressions in Trigonometry

In trigonometry, an expression is considered undefined when it involves division by zero. For example, the tangent function is undefined at angles where the cosine is zero, such as π/2 and 3π/2. Understanding when functions are undefined is crucial for accurately evaluating trigonometric expressions.