Problem 15 - Algebra and Trigonometry

Question

evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. $$ \cot \frac{\pi}{2} $$

Step-by-Step Solution

Answer

The cotangent of \(\frac{\pi}{2}\) is undefined.

Step 1: Understand the cotangent function

The cotangent of an angle is the reciprocal of the tangent of that angle. Therefore, \(\cot \theta = \frac{1}{\tan \theta}\).

Step 2: Evaluate tangent at the given angle

The tangent of \(\frac{\pi}{2}\) is undefined, because the sine function has a maximum value of 1 at \(\frac{\pi}{2}\), and the cosine function has a value of 0. The tangent is defined as the ratio of the sine to cosine, so at \(\frac{\pi}{2}\), \(\tan \frac{\pi}{2}\) becomes \(\frac{1}{0}\), which is undefined.

Step 3: Determine the value of the cotangent

As the cotangent function is the reciprocal of the tangent function, and the tangent of \(\frac{\pi}{2}\) is undefined, the cotangent of \(\frac{\pi}{2}\) is also undefined, because it is impossible to take the reciprocal of an undefined number.

Key Concepts

Cotangent

The cotangent function is one of the six fundamental trigonometric functions, like sine and cosine. It is the reciprocal of the tangent function. In mathematical terms, this is expressed as \( \cot \theta = \frac{1}{\tan \theta} \). When you hear "reciprocal," think of flipping fractions: if \( \tan \theta = \frac{a}{b} \), then \( \cot \theta = \frac{b}{a} \).

Understanding cotangent is crucial because it helps in analyzing angles and their relationships in various geometrical contexts. In standard unit circle terms, cotangent can be defined in terms of sine and cosine as \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

Keep in mind:

• If the tangent function is zero, the cotangent is undefined because you can't divide by zero.
• Cotangent is an odd function; it has the property \( \cot(-\theta) = -\cot(\theta) \).

Quadrantal Angles

Quadrantal angles are specific angles that lie along the x-axis or y-axis in the unit circle. Examples include angles such as 0, \( \frac{\pi}{2} \), \( \pi \), \( \frac{3\pi}{2} \), and so on.

These angles are uniquely significant because they occur at the boundaries between different quadrants and involve intersections with the axes. At these points:

• The sine, cosine, and tangent functions tend to reach their maximum, minimum, or undefined values.
• They often result in simple values such as 0, 1, or -1 for sine and cosine while making tangent functions zero or undefined.

When evaluating trigonometric functions at quadrantal angles, always pay attention to the value of the cosine and sine at these points. They can drastically change whether an expression becomes zero, one, or undefined.

Tangent Function

The tangent function connects two primary trigonometric functions — sine and cosine. It is defined as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Therefore, the value of the tangent depends fundamentally on sine and cosine values.

It's essential to remember some key properties of the tangent function:

• The tangent is periodic with a period of \( \pi \), meaning it repeats itself every \( \pi \) radians.
• It is undefined whenever \( \cos \theta = 0 \). This scenario occurs at odd multiples of \( \frac{\pi}{2} \) (like \( \frac{\pi}{2}, \frac{3\pi}{2} \)). This is because ratio by zero makes the value undefined.
• The function features vertical asymptotes where it is undefined.
• It has odd function symmetry, so \( \tan(-\theta) = -\tan(\theta) \).

Considering tangent's relationship to sine and cosine, it provides valuable insight into the behavior of angles and allows us to solve a variety of problems in trigonometry, particularly those involving right triangles.