Problem 4 - Precalculus

Question

In Exercises 1 - 20 , find the exact value or state that it is undefined. $$ \cot \left(\frac{4 \pi}{3}\right) $$

Step-by-Step Solution

Answer

\( \cot\left(\frac{4\pi}{3}\right) = \frac{\sqrt{3}}{3} \).

Step 1: Understand the Cotangent Function

The cotangent function, \( \cot(\theta) \), is defined as the reciprocal of the tangent function. Therefore, \( \cot(\theta) = \frac{1}{\tan(\theta)} \). Our task is to find \( \cot\left(\frac{4\pi}{3}\right) \).

Step 2: Find the Reference Angle

The angle \( \frac{4\pi}{3} \) radians is in the third quadrant. To find its reference angle, we subtract \( \pi \) from \( \frac{4\pi}{3} \):\[ \frac{4\pi}{3} - \pi = \frac{4\pi}{3} - \frac{3\pi}{3} = \frac{\pi}{3} \]. Thus, the reference angle is \( \frac{\pi}{3} \).

Step 3: Determine the Sign of Cotangent in Third Quadrant

In the third quadrant, the cotangent is positive because both sine and cosine are negative, leading \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \) to be positive.

Step 4: Calculate Cotangent of Reference Angle

For \( \frac{\pi}{3} \), we know that \( \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \), so its reciprocal is \( \cot\left(\frac{\pi}{3}\right) = \frac{1}{\sqrt{3}} \).

Step 5: Find Exact Value of \( \cot\left(\frac{4\pi}{3}\right) \)

As determined, the reference angle is \( \frac{\pi}{3} \) and cotangent is positive in the third quadrant. Therefore, \( \cot\left(\frac{4\pi}{3}\right) = \frac{1}{\sqrt{3}} \). To rationalize the denominator, we multiply by \( \frac{\sqrt{3}}{\sqrt{3}} \):\[ \frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3} \].

Key Concepts

Cotangent Function

To understand the cotangent function, let's start with its relationship to other trigonometric functions. Cotangent, denoted as \( \cot(\theta) \), is the reciprocal of the tangent function. If you know the tangent of an angle, you can find the cotangent by taking the inverse: \( \cot(\theta) = \frac{1}{\tan(\theta)} \). This makes it one of the basic trigonometric functions, alongside sine, cosine, and tangent.
One important aspect of cotangent is how it behaves in different quadrants of the unit circle. Unlike the cosine and sine functions that measure horizontal and vertical distances on the unit circle, the cotangent measures the ratio of the cosine to sine. This can be represented as \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \). It's crucial to remember this ratio since it tells us the nature of cotangent in various quadrants based on the signs of sine and cosine.

Reference Angle

When working with angles in trigonometry, especially those not in the first quadrant, reference angles become valuable. A reference angle is the acute angle that a given angle makes with the x-axis. No matter the original angle, the reference angle is always positive and less than \( \frac{\pi}{2} \) radians (or 90 degrees).
To determine the reference angle for \( \frac{4\pi}{3} \), we first need to understand its position in the unit circle. Since \( \frac{4\pi}{3} \) radians is greater than \( \pi \), it's located in the third quadrant. By subtracting \( \pi \) from \( \frac{4\pi}{3} \), we find the reference angle to be \( \frac{\pi}{3} \). Knowing how to compute the reference angle helps greatly in simplifying computations related to trigonometric functions.

Third Quadrant

Angles placed in the third quadrant hold special characteristics that influence trigonometric functions like cotangent. This quadrant is defined by angles that range from \( \pi \) to \( \frac{3\pi}{2} \) radians.
One key feature of the third quadrant is the sign of sine and cosine functions; both are negative here. However, because cotangent is the ratio \( \frac{\cos(\theta)}{\sin(\theta)} \), and both sine and cosine are negative, the cotangent itself becomes positive. Recognizing the sign of trigonometric functions in different quadrants is vital, as it helps to predict whether results will be positive or negative, thus simplifying calculations.

Reciprocal Function

The concept of reciprocal functions extends beyond just cotangent and tangent. Trigonometric functions relate to each other through reciprocal identities, each providing insight into different properties.
For trigonometric functions, here's a quick breakdown:

• Sine and cosecant are reciprocals: \( \csc(\theta) = \frac{1}{\sin(\theta)} \).
• Cosine and secant are reciprocals: \( \sec(\theta) = \frac{1}{\cos(\theta)} \).
• Tangent and cotangent are reciprocals: \( \cot(\theta) = \frac{1}{\tan(\theta)} \).

These identities help simplify calculations and solve trigonometric equations. Knowing that functions are reciprocals of one another means, you can convert between them effortlessly, like using \( \cot(\theta) \) when \( \tan(\theta) \) is known, allowing for a smoother calculation process.