Problem 62 - 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. $$\cot 1800^{\circ}$$

Step-by-Step Solution

Answer

The value of \( \cot 1800^\circ \) is undefined.

Step 1: Understanding the Point

To solve \(\cot 1800^\circ\), we first need to understand the position of the angle on the unit circle. Since angles on the unit circle repeat every \(360^\circ\), we simplify \(1800^\circ\) by finding the remainder when divided by \(360^\circ\). \[ 1800 \div 360 = 5 \, ext{remainder} \, 0 \]Thus, \(1800^\circ\) is equivalent to \(0^\circ\).

Step 2: Finding the Quadrantal Angle

With \(1800^\circ\) equivalent to \(0^\circ\), we locate the corresponding point on the unit circle. This is the positive x-axis, where the coordinates are \( (1, 0) \).

Step 3: Identifying Values of x, y, and r

At \(0^\circ\), the coordinates are \(x = 1\), \(y = 0\), and since it's on the unit circle, \(r = 1\). By definition, \\[ \cot \theta = \frac{x}{y} \]for a point \( (x, y)\) on the circle.

Step 4: Evaluating the Function

Substitute \(x = 1\) and \(y = 0\) into the cotangent formula: \[ \cot 0^\circ = \frac{1}{0} \] The expression is undefined because division by zero is undefined.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of one unit centered at the origin of a coordinate plane. Because the radius is one, any point

• along the circle is at a distance of 1 from the center.
• The equation of the unit circle is given by: \(x^2 + y^2 = 1\), which follows from the Pythagorean theorem.

Understanding the unit circle helps you visualize angles and trigonometric functions, as every point on the circle can be defined by

• its coordinates (x, y).

These coordinates correspond to the cosine and sine of the angle formed by the line connecting the origin to the point with the positive x-axis. For example, at \(0^\circ\), the circle intersects the positive x-axis at the point (1, 0), where \(\cos(0^\circ) = 1\) and \(\sin(0^\circ) = 0\).
Recognizing the importance of the unit circle will aid in understanding how we calculate trigonometric functions and angles.

Quadrantal Angles

Quadrantal angles are specific angles located at the intersection of the x- or y-axis and are a subset of the angles on the unit circle. They occur at regular intervals of 90 degrees or radians of \(\frac{\pi}{2}\). The important quadrantal angles are

• \(0^\circ\) (or \(360^\circ\)),
• \(90^\circ\),
• \(180^\circ\),
• and \(270^\circ\).

These angles are significant because they consistently show up in evaluations involving trigonometric functions.
For example, at \(180^\circ\), the point is on the negative x-axis with coordinates (-1, 0), and at \(90^\circ\), it sits on the positive y-axis with coordinates (0, 1). A simple way to remember quadrantal angles is that they are the "cornerstones" on the unit circle, positioned at the vertical and horizontal extremities.

Cotangent

The cotangent function, often abbreviated as \(\cot\), is a trigonometric function that is the reciprocal of the tangent function. It is defined as:

• \(\cot \theta = \frac{x}{y}\)

When \(\theta\) is the angle in standard position, \(x\) is the adjacent side, and \(y\) is the opposite side in trigonometry, but on the unit circle, \(x\) and \(y\) are simply the coordinates of the point.
For angles that fall on quadrants, such as \(0^\circ\), the cotangent becomes undefined. This is because at \(0^\circ\), the y-coordinate is 0, making the computation \(\cot 0^\circ = \frac{1}{0}\) indeterminate. This relationship between coordinates \(x\) and \(y\) is critical when dealing with trigonometric identities and values.

Undefined Expressions

In mathematics, expressions become undefined when they result in a situation that lacks meaning within the given context. For instance, division by zero does not yield a valid number, leading to an undefined expression.
For trigonometric functions such as cotangent, undefined expressions often occur with angles that align with quadrantal angles. This is because

• for certain angles, the denominator in the trigonometric formula becomes zero, resulting in an expression that cannot be computed further, such as \(\frac{x}{0}\).

Recognizing when a function is undefined helps in solving equations and avoiding potential errors in computation. For \(\cot 1800^\circ\), simplifying to \(\cot 0^\circ\) leads to an undefined expression, exemplifying why it's crucial to thoroughly understand the behavior of trigonometric functions.