Problem 159 - Calculus Volume 1

Question

For the following exercises, solve the trigonometric equations on the interval \(0 \leq \theta<2 \pi.\) $$\sqrt{3} \cot \theta+1=0$$

Step-by-Step Solution

Answer

The solutions are \( \theta = \frac{2\pi}{3} \) and \( \theta = \frac{5\pi}{3} \).

Step 1: Isolate the Cotangent Function

Start by isolating the cotangent function in the equation. The given equation is \( \sqrt{3} \cot \theta + 1 = 0 \). Subtract 1 from both sides to obtain \( \sqrt{3} \cot \theta = -1 \).

Step 2: Solve for Cotangent

Divide both sides of the equation by \( \sqrt{3} \) to solve for \( \cot \theta \). This gives \( \cot \theta = -\frac{1}{\sqrt{3}} \).

Step 3: Determine the Angle Using Known Values

Recall that \( \cot \theta = \frac{1}{\tan \theta} \), and \( \cot \theta = -\frac{1}{\sqrt{3}} \) implies \( \theta = \frac{2\pi}{3} \) or \( \theta = \frac{5\pi}{3} \) when using trigonometric identities and considering the range of \( \theta \).

Step 4: Verify Solutions in the Given Interval

Check the solutions to ensure they fall within the specified interval \( [0, 2\pi) \). Both \( \frac{2\pi}{3} \) and \( \frac{5\pi}{3} \) satisfy \( 0 \leq \theta < 2\pi \).

Key Concepts

Cotangent Function

The cotangent function, denoted as \( \cot \theta \), is a crucial concept in trigonometry. It is the reciprocal of the tangent function, meaning \( \cot \theta = \frac{1}{\tan \theta} \). This function has unique properties that make it useful for solving certain types of trigonometric equations.

• The cotangent function is undefined when \( \tan \theta = 0 \), which happens at angles like \( \theta = 0, \pi, 2\pi \).
• It is periodic with a period of \( \pi \), meaning it repeats its values after intervals of \( \pi \).
• The cotangent of an angle measures the ratio of the adjacent side to the opposite side in a right triangle, assuming the angle is not a right angle.

Understanding these properties helps when you need to manipulate or solve equations involving the cotangent function, as it allows you to predict behavior over different intervals.

Solving Equations

Solving trigonometric equations follows a systematic approach that simplifies the problem using basic algebraic operations and trigonometric principles. Here's an overview of the process:

• Isolate the Trigonometric Function: Begin by rearranging the equation to have the trigonometric function by itself on one side. This often involves basic operations such as addition, subtraction, multiplication, or division.


• Determine Possible Angles: Once isolated, find the specific angles that satisfy the trigonometric equation. For equations involving cotangent, remember that you're seeking angles where the cotangent takes specific values, keeping in mind the function's periodicity.
• Check the Interval: Finally, ensure that the angles you find fall within the problem's specified interval, such as \( 0 \leq \theta < 2\pi \). If needed, adjust by adding or subtracting intervals of the function's period.

These steps break down complex problems into more manageable parts, leading to successful solutions.

Trigonometric Identities

Trigonometric identities play an essential role in simplifying and solving equations. These identities express relationships between different trigonometric functions and can transform one form of an equation into another more solvable form. Here are some useful ones to know:

• Pythagorean Identities: These include \( \sin^2 \theta + \cos^2 \theta = 1 \), \( 1 + \tan^2 \theta = \sec^2 \theta \), and \( 1 + \cot^2 \theta = \csc^2 \theta \). They help relate different functions to each other.


• Reciprocal Identities: These state that \( \csc \theta = \frac{1}{\sin \theta} \), \( \sec \theta = \frac{1}{\cos \theta} \), and \( \cot \theta = \frac{1}{\tan \theta} \), which is crucial when manipulating the cotangent function.


• Angle Sum and Difference Identities: They allow the calculation of the sine, cosine, or tangent of an angle expressed as a sum or difference of two other angles, helping to solve specific trigonometric equations.

By mastering these identities, solving trigonometric equations becomes a process of application rather than trial and error, allowing for efficient problem solving.