Problem 199 - Precalculus

Question

For the following exercises, use reference angles to evaluate the expression. $$ \cot \frac{13 \pi}{6} $$

Step-by-Step Solution

Answer

The value of \( \cot \frac{13\pi}{6} \) is \( \sqrt{3} \).

Step 1: Identify the Quadrant

First, we need to determine the quadrant in which the angle \( \frac{13\pi}{6} \) lies. Start by converting \( \frac{13\pi}{6} \) into degrees if needed or evaluate it using its position in the radian circle. Since \( 2\pi \) is a full circle equivalent to \( \frac{12\pi}{6} \), \( \frac{13\pi}{6} \) is \( \frac{\pi}{6} \) more than \( 2\pi \). Since angles greater than \( 2\pi \) loop around the circle, \( \frac{13\pi}{6} \) corresponds to \( \frac{\pi}{6} \) in the first quadrant.

Step 2: Determine the Reference Angle

The reference angle is the smallest angle the terminal side of the given angle makes with the x-axis. Since \( \frac{13\pi}{6} \) gives us an effective angle of \( \frac{\pi}{6} \) (by subtracting \( 2\pi \)), the reference angle is \( \frac{\pi}{6} \).

Step 3: Evaluate the Cotangent of the Reference Angle

Now, we find \( \cot \frac{\pi}{6} \). Using the identity for cotangent, \( \cot \theta = \frac{1}{\tan \theta} \), and knowing that for \( \theta = \frac{\pi}{6}, \tan \frac{\pi}{6} = \frac{1}{\sqrt{3}} \). Therefore, \( \cot \frac{\pi}{6} = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3} \).

Step 4: Conclude with Result

The initial angle \( \frac{13\pi}{6} \) ends in the first quadrant, where the cotangent function remains positive. Hence, \( \cot \frac{13\pi}{6} = \sqrt{3} \).

Key Concepts

Reference Angles

Reference angles are a fundamental concept in trigonometry that help in simplifying the calculation of trigonometric functions for angles located in different quadrants. The reference angle is the smallest angle between the terminal side of the given angle and the x-axis. This angle is always positive and less than or equal to \(90^\circ\) or \( \frac{\pi}{2} \) radians.

To determine a reference angle:

• Identify the quadrant where the terminal side of the angle lies.
• Measure the smallest angle the terminal side forms with the x-axis.
• Use this reference angle to evaluate trigonometric functions for the original angle.

For instance, in the exercise, the reference angle of \( \frac{13\pi}{6} \) is found by understanding that beyond \(2\pi\), it resembles \( \frac{\pi}{6} \). Hence, \( \theta_{ref} = \frac{\pi}{6} \). This allows us to evaluate trigonometric functions using familiar angles.

Cotangent Function

The cotangent function, denoted as \( \cot \theta \), is the reciprocal of the tangent function. It is defined as \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \). Cotangent provides the ratio of the adjacent side to the opposite side of a right triangle.

Key properties of the cotangent function include:

• It is undefined when \( \theta \) is an integer multiple of \(\pi\) because \( \sin \theta = 0 \) at those points.
• \( \cot \theta \) is positive in the first and third quadrants, where both sine and cosine have the same sign.

In our exercise, finding \( \cot \frac{13\pi}{6} \) involved first reducing it to \( \cot \frac{\pi}{6} \), which is \( \sqrt{3} \), as it is a well-known cotangent value for reference angles.

Quadrants

Quadrants are divisions of the coordinate plane. Each quadrant corresponds to a specific combination of signs for the x (cosine) and y (sine) coordinates.

The first quadrant includes angles between \(0\) and \(\frac{\pi}{2}\). Both sine and cosine values are positive, making all trigonometric functions positive. In the second quadrant, only sine values are positive. The third quadrant allows for only the tangent and cotangent functions to remain positive.

The specific order of the quadrants is important for trigonometry:

• 1st Quadrant: \(0\) to \(\frac{\pi}{2}\)
• 2nd Quadrant: \(\frac{\pi}{2}\) to \(\pi\)
• 3rd Quadrant: \(\pi\) to \(\frac{3\pi}{2}\)
• 4th Quadrant: \(\frac{3\pi}{2}\) to \(2\pi\)

In the exercise, \( \frac{13\pi}{6} \) was reduced to \( \frac{\pi}{6} \), which is an angle in the first quadrant where the cotangent is positive.

Radian Measure

Radian measure is a way to express angles through the radius of a circle. One complete revolution around a circle is \(2\pi\) radians, indicating the connection between radians and the circle's circumference.

Understanding radian measure is crucial because:

• It allows for a monotonic and consistent way to measure angles.
• It establishes a basis for calculus-based trigonometry.
• It helps in simplifying calculations in trigonometric functions.

When working with radians, angles like \( \frac{13\pi}{6} \) indicate how many radii have wrapped around the circle. Simplifying these values using the periodic properties of trigonometric functions, allows us to compute functions like cotangent easily, which we demonstrated with \( \cot \frac{\pi}{6} \) returning \( \sqrt{3} \) as the solution.