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Chapter 5: Problem 10

Find the exact value of each of the following expressions without using a calculator. $$\cot (2 \pi / 3)$$

Short Answer

Expert verified
\(\cot(2\pi / 3) = -\frac{\sqrt{3}}{3}\)

Step by step solution

01

Understand the Angle

Identify that the angle given is in radians. The angle is \(2\pi / 3\). This is an angle in the second quadrant of the unit circle.
02

Find Reference Angle

A reference angle for \(2\pi / 3\) is found by subtracting \(\pi / 3\) from \(\pi \). Therefore, the reference angle is \(\pi / 3\).
03

Determine the Cotangent of the Reference Angle

\(\cot(\pi / 3) = \frac{1}{\tan(\pi / 3)}\). We know that \(\tan(\pi / 3) = \sqrt{3}\), so \(\cot(\pi / 3) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\).
04

Adjust for the Quadrant

Since \(2\pi / 3\) is in the second quadrant, and \(\cot\) is negative in the second quadrant, we have \(\cot(2\pi / 3) = -\cot(\pi / 3)\).
05

Compute the Final Answer

Using the value found in Step 3, we have \(\cot(2\pi / 3) = -\frac{\sqrt{3}}{3}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radians
Radians measure angles based on the radius of a circle. Unlike degrees which divide a circle into 360 parts, radians divide it using the circle's circumference. One full circle equals \(2\pi\) radians, which is about 6.28. This links arc length and radius directly. For example, a \(\pi/2\) radian means a quarter circle or 90 degrees. Converting between radians and degrees involves the factor \(\pi \,\text{radians} = 180 \degree\). Remembering common angles in radians鈥攍ike \(\pi/3\), \(\pi/4\), and \(2\pi/3\)鈥攎akes solving trigonometric problems easier.
Unit Circle
The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. It's fundamental in trigonometry. Each point on the unit circle corresponds to an angle and the (x, y) coordinates on the circle represent the cosine and sine of that angle, respectively. For instance, at \(\theta = 2\pi/3\), the point on the unit circle is \((-1/2, \sqrt{3}/2)\). This shows that \(\cos(2\pi/3) = -1/2\) and \(\sin(2\pi/3) = \sqrt{3}/2\). Understanding these coordinates helps identify values for all trigonometric functions, including cotangent.
Reference Angle
A reference angle is the acute angle formed between the terminal side of an angle and the x-axis. It's always positive and lies between \(0\) and \(\pi/2\) radians. For an angle \(2\pi/3\), subtract \(\pi/3\) from \(\pi\) to find the reference angle: \(\pi - 2\pi/3 = \pi/3\). By relating each angle to an easily known reference angle, we simplify our calculations. For example, \(\cot(\pi/3)\) is the same as \(\cot(2\pi/3)\) but with a sign change based on the quadrant.
Cotangent
Cotangent, written as \(\cot(\theta)\), is the reciprocal of tangent: \(\cot(\theta) = 1 / \tan(\theta)\). It's also calculated from the cosine and sine: \(\cot(\theta) = \cos(\theta) / \sin(\theta)\). For \(\theta = \pi/3\), \(\tan(\pi/3) = \sqrt{3}\), so \(\cot(\pi/3) = 1/\sqrt{3} = \sqrt{3}/3\). Remember that the sign of cotangent depends on the angle's quadrant. In the second quadrant, where \(2\pi/3\) lies, cotangent values are negative, hence \(\cot(2\pi/3) = -\cot(\pi/3) = -\sqrt{3}/3\).

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