Problem 16 - Algebra and Trigonometry

Question

Find the exact value of the trigonometric function at the given real number. $$ \text { (a) }\cot \left(-\frac{\pi}{3}\right) \quad \text { (b) } \cot \frac{2 \pi}{3} \quad \text { (c) } \cot \frac{5 \pi}{3} $$

Step-by-Step Solution

Answer

(a) \(-\frac{1}{\sqrt{3}}\); (b) \(-\frac{1}{\sqrt{3}}\); (c) \(-\frac{1}{\sqrt{3}}\).

Step 1: Understanding Cotangent Function

The cotangent function, \( \cot \theta \), is the reciprocal of the tangent function, \( \tan \theta \). It is defined as \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \). We will use this definition to find the exact values for each angle.

Step 2: Evaluate \( \cot \left(-\frac{\pi}{3}\right) \)

First, recognize the angle \( -\frac{\pi}{3} \). Since the cotangent function is an odd function (i.e., \( \cot(-\theta) = -\cot(\theta) \)), we have \( \cot\left(-\frac{\pi}{3}\right) = -\cot\left(\frac{\pi}{3}\right) \). From trigonometric identities, we know \( \cot\left(\frac{\pi}{3}\right) = \frac{1}{\sqrt{3}} \), so \( \cot\left(-\frac{\pi}{3}\right) = -\frac{1}{\sqrt{3}} \).

Step 3: Evaluate \( \cot \frac{2\pi}{3} \)

The angle \( \frac{2\pi}{3} \) lies in the second quadrant, where cosine is negative and sine is positive. In this quadrant, \( \cot \frac{2\pi}{3} = \frac{\cos \frac{2\pi}{3}}{\sin \frac{2\pi}{3}} = \frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} \).

Step 4: Evaluate \( \cot \frac{5\pi}{3} \)

The angle \( \frac{5\pi}{3} \) is equivalent to \( \frac{-\pi}{3} + 2\pi \), which is in the fourth quadrant. In this quadrant, cosine is positive and sine is negative. \( \cot \frac{5\pi}{3} = \frac{\cos \frac{5\pi}{3}}{\sin \frac{5\pi}{3}} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} \).

Step 5: Compile Results

For the three parts given, the exact values are: (a) \( \cot \left( -\frac{\pi}{3} \right) = -\frac{1}{\sqrt{3}} \), (b) \( \cot \frac{2\pi}{3} = -\frac{1}{\sqrt{3}} \), and (c) \( \cot \frac{5\pi}{3} = -\frac{1}{\sqrt{3}} \).

Key Concepts

Cotangent Function

The cotangent function is one of the six fundamental trigonometric functions, primarily used in geometry, calculus, and engineering. The cotangent of an angle, denoted as \( \cot \theta \), is the reciprocal of the tangent function. This can be expressed as \( \cot \theta = \frac{1}{\tan \theta} \). Additionally, it can be represented in terms of sine and cosine: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).
This function is undefined whenever the sine of the angle is zero, as division by zero is undefined in mathematics. The cotangent function is periodic, with a period of \( \pi \), meaning the function repeats its values every \( \pi \) units.
Understanding the periodicity and behavior of the cotangent function in different quadrants is crucial, as it helps us understand the signs and values in diverse applications.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for any angle. They are essential tools for solving trigonometric equations and simplifying expressions.
Some of the most fundamental trigonometric identities involving the cotangent function include:

• The reciprocal identity: \( \cot \theta = \frac{1}{\tan \theta} \)
• The quotient identity: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)
• The Pythagorean identity related to cotangent: \( 1 + \cot^2 \theta = \csc^2 \theta \)

These identities are used to transform expressions and solve equations more efficiently. The odd-function identity, \( \cot(-\theta) = -\cot(\theta) \), is particularly useful when dealing with negative angles, as it demonstrates symmetry about the origin in the trigonometric graph.

Quadrant Angles

In trigonometry, the angle's position within the Cartesian coordinate system determines the sign of its trigonometric functions. The coordinate plane is divided into four quadrants that are useful for understanding the behavior of functions like sine, cosine, and cotangent.
- In the **first quadrant**: All trigonometric functions are positive.
- In the **second quadrant**: Sine is positive, but cosine and cotangent are negative.
- In the **third quadrant**: Tangent and cotangent are positive, while sine and cosine are negative.
- In the **fourth quadrant**: Cosine is positive, but sine and cosine are negative.
Quadrant angles help to determine the sign of the trigonometric values easily. For example, the angle \( \frac{2\pi}{3} \) is in the second quadrant where \( \cot \) is negative, whereas \( \frac{5\pi}{3} \) lies in the fourth quadrant with the same negative sign for \( \cot \). This understanding assists in evaluating angles correctly without having to plot them.