Problem 51 - A Graphical Approach to Precalculus with Limits

Question

Begin by reproducing the graph in RiGuRE as. Keep in mind that for each of the four points labeled in the figure, \(r=1 .\) For each quadrantal angle, identify the appropriate values of \(x, y,\) and \(r\) to find the indicated function value. If it is undefined, say so. Check your answers with a calculator in degree mode. $$\text { sec } 180^{\circ}$$

Step-by-Step Solution

Answer

\(\sec(180^{\circ}) = -1\).

Step 1: Understanding the Problem

We need to find the value of the secant function (\(\sec\)) for the angle \(180^{\circ}\). Remember that the secant function is the reciprocal of the cosine function, so \(\sec(\theta) = \frac{1}{\cos(\theta)}\).

Step 2: Identify Coordinates at Angle

For \(180^{\circ}\), consider the point on the unit circle. For this angle, the corresponding point is \((-1, 0)\), where \(x = -1\), \(y = 0\), and \(r = 1\) because \(r = 1\) for any point on the unit circle.

Step 3: Find Cosine Value

The cosine of an angle in the unit circle is the \(x\)-coordinate of the corresponding point. Therefore, \(\cos(180^{\circ}) = -1\).

Step 4: Calculate Secant Value

Since \(\sec(\theta) = \frac{1}{\cos(\theta)}\), substitute \(\cos(180^{\circ}) = -1\) into the formula. This gives \(\sec(180^{\circ}) = \frac{1}{-1} = -1\).

Step 5: Verification Using Calculator

Use a calculator set in degree mode and find \(\sec(180^{\circ})\). Confirm that \(\sec(180^{\circ}) = -1\). This should match our calculated value.

Key Concepts

Unit Circle

The Unit Circle is a critical tool in trigonometry, representing a circle with a radius of 1, centered at the origin \(0, 0\) on a coordinate plane. This simple geometric concept is powerful as it helps decipher the trigonometric functions of different angles. Every point on the unit circle can be described using coordinates \(x, y\), where \(x\) represents the cosine of the angle and \(y\) represents the sine of the angle.

Because the radius \(r\) is always 1, this set-up makes it easy to calculate the values of trigonometric functions for specific angles. To illustrate, if you picture the x-axis and y-axis intersecting the circle, they do so at key points (1,0), (0,1), (-1,0), and (0,-1). Each of these represents what we call quadrantal angles. The value of coordinates on these angles helps solve complex trigonometry problems.

Secant Function

The Secant Function is one of the reciprocal trigonometric functions, specifically the reciprocal of the Cosine Function. This means, for an angle \( \theta \), the secant is defined as \( \sec(\theta) = \frac{1}{\cos(\theta)} \).

It displays the same periodic behavior that characterizes all trigonometric functions, although it has its own distinct graph that looks quite different from a sine or cosine graph. When the cosine of an angle equals zero, the secant is undefined because division by zero is impossible. In these instances, the secant function will have vertical asymptotes.

Understanding how to work with secant can be especially important when solving problems involving reciprocals of circular functions, such as angle and side calculations in trigonometry.

Quadrantal Angles

Quadrantal Angles are the angles situated on the axes of the unit circle. These angles include \(0^\circ, 90^\circ, 180^\circ,\) and \(270^\circ\). For these angles, one of the coordinates is always zero due to their position right on the x or y-axis.

For example:

• At \(0^\circ\), the point is \( (1, 0) \)
• At \(90^\circ\), it is \( (0, 1) \)
• At \(180^\circ\), it is \((-1, 0) \)
• At \(270^\circ\), it is \( (0, -1) \)

These coordinates are particularly helpful in identifying trigonometric function values for quadrantal angles as you only need to observe the coordinates relative to the axes. The calculations are straightforward, often simplifying into integers or zero.

Cosine Function

The Cosine Function is a fundamental trigonometric function that relates the angle in a right triangle to the lengths of the adjacent side and the hypotenuse or, in the unit circle, the x-coordinate of a point. In mathematical terms, for any angle \( \theta \), the cosine is expressed as \( \cos(\theta) \).

On the unit circle, this makes determining cosine values straightforward since they directly correspond to the x-coordinates of the circle's circumference. For instance, at \(0^\circ\), \(\cos(0^\circ) = 1\); at \(90^\circ\), \(\cos(90^\circ) = 0\); at \(180^\circ\), \(\cos(180^\circ) = -1\); and at \(270^\circ\), \(\cos(270^\circ) = 0\).

The cosine function also shares its periodic properties with the sinusoidal functions, repeating every \(360^\circ\) or \(2\pi\), which makes it predictable and extremely useful when analyzing waveforms or periodic phenomena.