Problem 11 - Precalculus Essentials

Question

In Exercises \(9-16\), evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. $$\sec \pi$$

Step-by-Step Solution

Answer

The value of \(\sec(\pi)\) is -1.

Step 1: Identify the Quadrantal Angle

The given angle is \(\pi\). In an unit circle, the angle \(\pi\) is associated with the point (-1, 0).

Step 2: Determine the Corresponding Cosine Value

At the angle of \(\pi\), the cosine value is calculated by considering the x-coordinate of the specified point on the unit circle. So, \(\cos(\pi) = -1\)

Step 3: Calculate the Secant Value

The secant function is the reciprocal of the cosine function. Therefore, the value of the secant function for the given angle can be computed as \(\sec(\pi) = 1/\cos(\pi) = 1/-1 = -1\)

Key Concepts

Understanding Quadrantal Angles

Quadrantal angles are special angles located at the axes of the unit circle. These angles are multiples of \(\frac{\pi}{2}\) radians or 90 degrees. They coincide with the x and y axes at specific points.
Understanding these angles is crucial because they determine the sine and cosine values as either -1, 0, or 1. The most common quadrantal angles on the unit circle include:

• 0 or 0 \(\pi\) (point (1, 0))
• \(\frac{\pi}{2}\) or 90° (point (0, 1))
• \(\pi\) or 180° (point (-1, 0))
• \(\frac{3\pi}{2}\) or 270° (point (0, -1))

Each of these points helps identify the sine and cosine effectively, making it easier to work with trigonometric functions. Especially when evaluating functions like cosine and secant, being familiar with where these angles lie on the unit circle simplifies the process greatly.

Exploring the Unit Circle

The unit circle is a crucial concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. The unit circle is useful in defining trigonometric functions for all real numbers. Like any circle, it can be described by the equation:\[x^2 + y^2 = 1\]The primary utility of the unit circle is to provide a straightforward way to determine the values of sine and cosine. It associates every angle, expressed in radians, with a specific (x, y) point on the circle.
These coordinates represent the cosine and sine of the angle, respectively:

• The x-coordinate is the cosine of the angle.
• The y-coordinate is the sine of the angle.

For example, at the angle \(\pi\) radians, the point on the unit circle is (-1, 0). Thus, \(\cos(\pi) = -1\) and \(\sin(\pi) = 0\). The understanding of the unit circle makes it easier to tackle problems involving quadrantal angles or any standard trigonometric evaluation.

Diving into the Cosine Function

The cosine function relates the angle of a right triangle to the ratio of the adjacent side over the hypotenuse. On the unit circle, cosine takes a role as the x-coordinate of the point corresponding to a given angle. More intuitively, it indicates how far along the horizontal axis from the origin a point lies.
**Key Properties of the Cosine Function**

• **Periodicity:** Cosine has a repeating interval of \(2\pi\) radians.
• **Range:** The range spans from -1 to 1.
• **Symmetry:** The cosine function is even, demonstrating symmetry about the y-axis by the identity \(\cos(-\theta) = \cos(\theta)\).

Understanding cosine in terms of the unit circle allows you to easily recognize key cosine values at quadrantal angles, such as 0, \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\). For instance, \(\cos(\pi) = -1\), a fact used to compute \(\sec(\pi)\) as a reciprocal.

Introduction to the Secant Function

The secant function is a lesser-known but important trigonometric function, defined as the reciprocal of the cosine function. In symbolic terms, for a given angle \(\theta\), it is expressed as: \[ \sec(\theta) = \frac{1}{\cos(\theta)} \]**Properties of the Secant Function**

• **Undefined Values:** Secant is undefined for angles where cosine equals zero, commonly at points \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\).
• **Range:** Unlike cosine, the range of secant excludes the interval (-1, 1), instead reaching from (-∞, -1] ∪ [1, ∞).

In solving our particular exercise, \(\sec(\pi)\) involved taking the reciprocal of \(\cos(\pi)\). Since \(\cos(\pi) = -1\), it follows that \(\sec(\pi) = \frac{1}{-1} = -1\). Recognizing such relationships through the properties of secant function gives insights into evaluating it precisely for any angle on the unit circle.