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Chapter 6: Problem 14

Find the exact solutions of the given equations, in radians. $$\cot x-\sqrt{3}=0$$

Short Answer

Expert verified
The exact solutions of the equation are \(x = \pi/3\) and \(x = 4\pi/3\) in radians.

Step by step solution

01

Isolate the cotangent term

To start solving the equation \(\cot x - \sqrt{3} = 0\), first isolate the cotangent term. Adjust the equation to yield: \(\cot x = \sqrt{3}\)
02

Use cotangent property

The cotangent of an angle in an equilateral triangle is \(\sqrt{3}\) at \(60掳\) or \(\pi/3\) radians. It is also \(\sqrt{3}\) at \(180掳+60掳=240掳\) or \(4\pi/3\) radians. Therefore, the solutions for x are \(\pi/3\) and \(4\pi/3\) within one cycle.
03

Find all solutions in radians within the range of 0 to \(2\pi\)

The solutions for x which satisfies the equation are: x = \(\pi/3 + k\pi\) and x = \(4\pi/3 + k\pi\), where k is an integer. However, since it is only asked for the solutions within the range of 0 to \(2\pi\) and cotangent has a period of \(\pi\), we only consider x = \(\pi/3\) and x = \(4\pi/3\) in radians.

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Key Concepts

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

Cotangent Function
The cotangent function, abbreviated as \( \cot \), is one of the fundamental trigonometric functions. It is defined as the reciprocal of the tangent function or the ratio of the adjacent side to the opposite side in a right triangle when angle \( x \) is at the vertex.
\[\cot x = \frac{1}{\tan x} = \frac{\text{adjacent}}{\text{opposite}}\]
In the context of a unit circle, \( \cot x \) can be expressed as \( \frac{\cos x}{\sin x} \). It is important to note that the cotangent function is undefined wherever the tangent function is zero.
  • The function is periodic, with a period of \( \pi \).
  • It has vertical asymptotes at intervals of \( k\pi \), where \( k \) is an integer.
  • It decreases continuously over each period moving from \( \infty \) to \(-\infty \).
Understanding the behavior of the cotangent function is crucial for solving equations that involve it.
Exact Solutions
Finding the exact solutions of trigonometric equations means determining the specific angles that satisfy the given equation. When dealing with trigonometric equations such as \( \cot x = \sqrt{3} \), exact values are found at angles whose trigonometric ratios are well-known.
  • For many trigonometric problems, angles like \( \pi/3 \), \( \pi/4 \), \( \pi/6 \), etc., have exact trigonometric values derived from the properties of special triangles or the unit circle.
  • Exact solutions indicate specific points on the trigonometric circle corresponding to those angles.
  • Unlike decimal approximations, the exact values are derived directly from fundamental geometric relationships and identities.
In our exercise, the problem gives us \( \cot x = \sqrt{3} \). We then look for angles where this condition is inherently true, resulting in solutions at \( \pi/3 \) and \( 4\pi/3 \) within one cycle of the cotangent's periodicity, \( \pi \).
Radians Measure
The radian is the standard unit of angular measure used in many areas of mathematics. Unlike degrees, which split a circle into 360 parts, radians are based on the radius length; specifically, a circle's circumference divided by its radius is approximately \( 2\pi \), linking \( 2\pi \) radians to a full circle.
  • One complete revolution around a circle corresponds to \( 2\pi \) radians.
  • Common angles in radians include \( \pi/2 \), \( \pi \), \( 3\pi/2 \), etc.
  • To convert degrees to radians, use the formula \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \).
In questions like ours, where solutions are sought in radians, the solutions are inherently tied to the unit circle and its properties. Thus, understanding radians facilitates precision and simplifies the handling of trigonometric equations in mathematical analysis.

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