Problem 55 - 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. $$\text { cot } 540^{\circ}$$

Step-by-Step Solution

Answer

The function value \(\text{cot}(540^{\circ})\) is undefined.

Step 1: Understand cotangent function

The cotangent function, cot(θ), is the reciprocal of the tangent function. For a given angle θ, \(\text{cot}(\theta) = \frac{1}{\text{tan}(\theta)}\). It is undefined wherever the tangent is 0.

Step 2: Identify the angle on the unit circle

The angle \(540^{\circ}\) is greater than \(360^{\circ}\) so we need to find its equivalent angle between \(0^{\circ}\) and \(360^{\circ}\). Subtract \(360^{\circ}\) from \(540^{\circ}\) to get \(540^{\circ} - 360^{\circ} = 180^{\circ}\). So, \(540^{\circ}\) is equivalent to \(180^{\circ}\).

Step 3: Find coordinates of the equivalent angle

At \(180^{\circ}\), the coordinates on the unit circle are \((-1, 0)\). Here, \(x = -1\), \(y = 0\), and the radius \(r = 1\).

Step 4: Calculate the tangent of the angle

Since \(\text{tan}(\theta) = \frac{y}{x}\), the tangent of \(180^{\circ}\) is \(\frac{0}{-1} = 0\).

Step 5: Evaluate the cotangent function

The cotangent is the reciprocal of the tangent. Therefore, \(\text{cot}(180^{\circ}) = \frac{1}{0}\). The cotangent is undefined because division by zero is undefined.

Key Concepts

Cotangent

The cotangent function is an essential trigonometric ratio that is the reciprocal of the tangent function. For a given angle \( \theta \), the cotangent is expressed as \( \text{cot}(\theta) = \frac{1}{\text{tan}(\theta)} \).
This means that the cotangent value is undefined wherever the tangent of \( \theta \) equals zero because division by zero is mathematically undefined.

• If \( \text{tan}(\theta) = 0 \), then \( \text{cot}(\theta) \) does not exist.
• The cotangent function is most frequently used in trigonometric calculations involving angles and their properties.

Knowing how and when the cotangent becomes undefined helps in understanding graphs and solving trigonometric equations.

Unit Circle

The unit circle is a circle with a radius of one unit centered at the origin of the coordinate plane. This fundamental concept is important for understanding trigonometric functions.

• Every point on the unit circle represents a pair \((x, y)\) and the radius \( r = 1 \), which is intrinsic for calculating angles and their sine, cosine, and tangent values.
• The quadrantal angles such as \(0^{\circ}, 90^{\circ}, 180^{\circ}, 270^{\circ}, \text{and} \ 360^{\circ}\) are key points on the unit circle.
• The coordinates of these specific angles form the basis for trigonometric ratios.

By representing angles as points on the unit circle, it becomes easier to visualize and compute their trigonometric functions, like cotangent and tangent.

Tangent

The tangent function, \( \text{tan}(\theta) \), represents the ratio of the opposite side to the adjacent side in a right-angled triangle. On the unit circle, it is defined as \( \text{tan}(\theta) = \frac{y}{x} \).

• It has values which repeat over intervals of \( 180^{\circ} \), meaning that the tangent function has a periodicity that aligns with these intervals.
• The points where tangent equals zero are crucial as they determine where cotangent becomes undefined.
• Understanding the behavior of tangent aids greatly in grasping its reciprocal function, the cotangent.

For instance, for an angle of \(180^{\circ}\), \( x = -1 \) and \( y = 0 \) on the unit circle, which makes \( \text{tan}(180^{\circ}) = 0 \). This causes the cotangent at this angle to be undefined.

Angle Equivalence

Angle equivalence refers to finding an equivalent angle within the primary interval of \(0^{\circ}\) to \(360^{\circ}\).

• For any angle \( \theta \) larger than \(360^{\circ}\), you can find its equivalent by subtracting multiples of \(360^{\circ}\).
• This process helps in reducing complex angles to simpler, standard angles for easier computation.
• In trigonometry, expressing angles in their equivalent form within this interval is crucial for simplification and calculation of trigonometric values.

In our exercise, \(540^{\circ}\) is reduced by subtracting \(360^{\circ}\), which results in an equivalent angle of \(180^{\circ}\). This simplifies evaluating the trigonometric functions and their corresponding coordinates on the unit circle.