Problem 187 - Precalculus

Question

For the following exercises, find the exact value of each expression. $$ \cot \frac{\pi}{6} $$

Step-by-Step Solution

Answer

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

Step 1: Understand Cotangent Function

The cotangent of an angle is the reciprocal of the tangent of that angle. Mathematically, it can be expressed as \( \cot \theta = \frac{1}{\tan \theta} \).

Step 2: Calculate \( \tan \frac{\pi}{6} \)

Using the unit circle or known trigonometric values, \( \tan \frac{\pi}{6} \) is known to be \( \frac{1}{\sqrt{3}} \) or \( \sqrt{3}/3 \).

Step 3: Find Reciprocal of \( \tan \frac{\pi}{6} \)

Since \( \cot \frac{\pi}{6} = \frac{1}{\tan \frac{\pi}{6}} \), calculate \( \frac{1}{\tan \frac{\pi}{6}} = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3} \).

Step 4: State the Exact Value

Now that we have the reciprocal, the exact value of \( \cot \frac{\pi}{6} \) is \( \sqrt{3} \).

Key Concepts

Cotangent Function

The cotangent function is an essential part of trigonometry. It is related to the angle of a right triangle, similar to sine, cosine, and tangent. In simple terms, the cotangent of an angle is the reciprocal of the tangent of that angle. This means that if you know the tangent, you can easily calculate the cotangent by flipping the tangent value. Mathematically, it is represented as:

• \( \cot \theta = \frac{1}{\tan \theta} \)

The cotangent function is denoted by \( \cot \), and it's helpful for solving various trigonometric problems. For example, when you have the tangent value, finding the cotangent is straightforward. Remember, while dealing with angles in trigonometry, you will often switch between these related functions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are always true for any angle, and they help make calculations easier. The connection between cotangent and tangent is one such identity. Additionally, identities like Pythagorean identities support converting and simplifying expressions.

Here are a few key identities involving cotangent:

• \( \cot^2 \theta + 1 = \csc^2 \theta \) (Pythagorean identity)
• \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

These identities are incredibly useful when working with angles and trigonometric equations. They allow you to express trigonometric functions in different forms, aiding in simplification and solution of problems. Knowing these identities can make trigonometry less daunting and more intuitive.

Unit Circle

The unit circle is a vital tool in understanding trigonometry. It is a circle with a radius of 1, positioned on the coordinate plane centered at the origin

• (0, 0). The equations of the circle are directly connected to the trigonometric functions such as sine, cosine, and tangent.

On the unit circle, each angle corresponds to a point with coordinates \((\cos \theta, \sin \theta)\). This makes it easy to evaluate trigonometric values of common angles by simply looking at the coordinates of these points. For example, the angle of \( \frac{\pi}{6} \) on the unit circle corresponds to the point \((\frac{\sqrt{3}}{2}, \frac{1}{2})\), from which we derive values like \( \tan \frac{\pi}{6} \).

Through the unit circle, you can visualize the behavior of trigonometric functions and understand the cyclic nature of these functions. It's a powerful representation that simplifies the way we look at angles and trigonometric relationships.