Problem 45 - Algebra and Trigonometry

Question

Find all solutions of the equation in the interval \([0,2 \pi).\) $$\tan x-3 \cot x=0$$

Step-by-Step Solution

Answer

The solutions are \( \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3} \).

Step 1: Express Cotangent in terms of Tangent

Recall that the cotangent function can be written as the reciprocal of the tangent function: \( \cot x = \frac{1}{\tan x} \). Substitute \( \cot x = \frac{1}{\tan x} \) in the equation \( \tan x - 3 \cot x = 0 \) to get \( \tan x - 3 \left(\frac{1}{\tan x}\right) = 0 \).

Step 2: Eliminate fractions by multiplying through by \( \tan x \)

Multiply the entire equation by \( \tan x \) to eliminate the fraction: \( \tan^2 x - 3 = 0 \). This simplifies the problem to solving a quadratic equation in terms of \( \tan x \).

Step 3: Solve the Quadratic Equation

Set \( \tan^2 x = 3 \). Taking the square root of both sides, we have \( \tan x = \pm \sqrt{3} \).

Step 4: Find Angles Corresponding to \( \tan x = \sqrt{3} \) and \( \tan x = -\sqrt{3} \)

For \( \tan x = \sqrt{3} \), the reference angle is \( \frac{\pi}{3} \). This happens in the first and third quadrants within \([0, 2\pi) \), so the solutions are \( x = \frac{\pi}{3} \) or \( x = \frac{4\pi}{3} \).For \( \tan x = -\sqrt{3} \), the reference angle is \(-\frac{\pi}{3} \). This occurs in the second and fourth quadrants, so \( x = \frac{2\pi}{3} \) or \( x = \frac{5\pi}{3} \).

Step 5: List All Solutions in the Interval \([0, 2\pi)\)

Gathering all the solutions from Step 4, the solutions to the equation \( \tan x - 3 \cot x = 0 \) in the interval \( [0, 2 \pi) \) are \( \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3} \).

Key Concepts

Tangent Function

The tangent function, commonly denoted as \( \tan x \), is a fundamental trigonometric function arising from the ratio of the sine and cosine functions. Specifically, it is defined by the equation \( \tan x = \frac{\sin x}{\cos x} \). This means that the value of tangent depends on the values of sine and cosine for the angle \( x \).

Tangent has a periodicity of \( \pi \), meaning it repeats its values every \( \pi \) radians. One unique feature of the tangent function is its vertical asymptotes, which occur whenever \( \cos x = 0 \), as the division by zero is undefined. This happens at odd multiples of \( \frac{\pi}{2} \) radians. Consequently, the graph of the tangent function displays repeating interesting behavior that increases and decreases rapidly around these points.

• Range: All real numbers \((-\infty, \infty)\).
• Period: \( \pi \).
• Zeroes: Occur at integer multiples of \( \pi \).

Understanding these properties is vital for manipulating and solving trigonometric equations involving the tangent function, such as finding reference angles or transforming expressions.

Cotangent Function

The cotangent function, denoted as \( \cot x \), is defined as the reciprocal of the tangent function. Mathematically, it is expressed as \( \cot x = \frac{1}{\tan x} \) or equivalently, \( \cot x = \frac{\cos x}{\sin x} \). This relationship shows that when \( \tan x \) is undefined, \( \cot x \) takes advantage of the sine portion instead of cosine.

Similar to the tangent, cotangent also has a periodicity, repeating every \( \pi \) radians. However, its behavior differs; it has vertical asymptotes wherever \( \sin x = 0 \), i.e., at integer multiples of \( \pi \). This contrasts with tangent's undefined points, making cotangent a complementary function to tangent in various mathematical analyses.

• Range: All real numbers \((-\infty, \infty)\).
• Period: \( \pi \).
• Zeroes: Occur at odd multiples of \( \frac{\pi}{2} \).

The cotangent function's distinct properties make it an intriguing and essential part of solving equations where the function roles are interchangeable.

Quadratic Equations

Quadratic equations are algebraic expressions of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). In the context of trigonometric equations like \( \tan^2 x - 3 = 0 \), we consider \( \tan x \) as analogous to the typical variable used in algebra, which allows us to solve them similarly.

The standard approach involves rearranging the expression into \( \tan^2 x = 3 \), simplifying to \( \tan x = \pm \sqrt{3} \) after taking square roots. This transition from quadratic to linear problem-solving is a common technique in mathematics.

• Standard form: \( ax^2 + bx + c = 0 \)
• Solution approach: Factoring or applying the quadratic formula, when necessary.
• Roots can be real or complex depending on the discriminant \( b^2 - 4ac \)

Coupling these algebraic methods with trigonometric substitutions provides practical insights into resolving complex expressions with trigonometric functions.

Reference Angles

Reference angles are a handy concept in trigonometry, used to simplify the analysis of trigonometric functions in any of the four quadrants. The reference angle is the smallest angle between the terminal side of the given angle and the x-axis.

For instance, when solving \( \tan x = \sqrt{3} \), the reference angle is \( \frac{\pi}{3} \). This indicates where \( \tan x \) naturally equals \( \sqrt{3} \)—in this case, the angle \( \theta = \frac{\pi}{3} \) and its symmetric counterpart in the third quadrant, \( \theta = \frac{4\pi}{3} \).

• Located in the first quadrant: Always positive
• Correlates with angles in all quadrants.
• Aids in visualizing solutions within a given interval.

Understanding reference angles enables students to recognize corresponding trigonometric values from one quadrant to another, establishing a framework for tackling problems across various angles.