Problem 72 - Calculus for Biology and Medicine

Question

Solve \(\sec ^{2} x=\sqrt{3} \tan x+1\) on \([0, \pi)\).

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = \frac{\pi}{3}\).

Step 1: Substitute Trigonometric Identities

We start by using the identity for secant squared and tangent: \(\sec^2 x = 1 + \tan^2 x\). This gives us the equation \(1 + \tan^2 x = \sqrt{3} \tan x + 1\).

Step 2: Simplify the Equation

Subtract 1 from both sides to remove the constant term, leading to the equation \(\tan^2 x = \sqrt{3} \tan x\).

Step 3: Factor the Equation

We can factor the equation as \(\tan x (\tan x - \sqrt{3}) = 0\). This implies two scenarios: either \(\tan x = 0\) or \(\tan x = \sqrt{3}\).

Step 4: Solve for \(\tan x = 0\)

We find \(\tan x = 0\) at \(x = 0\) and \(x = \pi\). Since the interval is \([0, \pi)\), only \(x = 0\) is included.

Step 5: Solve for \(\tan x = \sqrt{3}\)

This occurs when \(x = \frac{\pi}{3}\) and its periodic equivalent. Within the interval \([0, \pi)\), this corresponds to \(x = \frac{\pi}{3}\).

Step 6: Combine Solutions

The solutions for \(x\) within the interval \([0, \pi)\) are \(x = 0\) and \(x = \frac{\pi}{3}\).

Key Concepts

Secant Function

The secant function, denoted as \( \sec x \), is the reciprocal of the cosine function. Mathematically, it is expressed as \( \sec x = \frac{1}{\cos x} \). Understanding the secant function is critical to solving trigonometric equations, as it often appears in problems involving angles and their relationships. When dealing with secant, it is important to remember:

• The secant function is undefined wherever the cosine is zero because division by zero is undefined.
• Characteristics: It has the same period and vertical asymptotes as the tangent function, occurring where \( \cos x = 0 \).
• Graphically, the secant function exhibits vertical "spikes" at each of these asymptotes and fluctuates above and below the x-axis.

Utilizing the secant identity \( \sec^2 x = 1 + \tan^2 x \), we can transform equations to facilitate their solutions by connecting it with the tangent function.

Tangent Function

The tangent function, represented as \( \tan x \), relates to sine and cosine through the formula \( \tan x = \frac{\sin x}{\cos x} \). It is crucial to comprehend this function in trigonometric equations because:

• The tangent function describes the slope of the angle formed with the x-axis.
• It has periodic properties, repeating every \( \pi \) radians. This means any solution within a cycle can be repeated for other cycles.
• There are vertical asymptotes wherever the cosine function is zero, similar to the secant function.

Its role in the exercise is pivotal because simplifying equations often involves transforming multiple trigonometric expressions to tangent.

Factoring Equations

Factoring is a powerful algebraic tool used to solve equations efficiently. In the context of trigonometric equations, it involves expressing a polynomial equation in terms of its factors. This method simplifies the equation, allowing us to identify solutions quickly. Here is why factoring is useful:

• It breaks down complex expressions into simpler components, helping identify possible solutions.
• By setting each factor equal to zero, it offers a clear path to finding the roots of the equation.
• In trigonometric problems, factoring can reveal specific angles or values that solve the equation, as seen in our step-by-step solution using \( \tan x (\tan x - \sqrt{3}) = 0 \).

Using factoring, we separate the equation into manageable parts, simplifying the solving process for values \( x = 0 \) and \( x = \frac{\pi}{3} \).

Trigonometric Identities

Trigonometric identities are formulas that express relationships between the angles and sides of a triangle. They are immensely useful in simplifying trigonometric equations, allowing for straightforward solutions. Key identities include:

• Pythagorean identities: Used in this exercise is \( \sec^2 x = 1 + \tan^2 x \), which connects secant and tangent.
• Reciprocal identities: These include \( \tan x = \frac{\sin x}{\cos x} \), essential for simplifying expressions.
• Angle-sum identities: Though not directly used here, these assist in combining angles efficiently in more complex problems.

Utilizing these identities can significantly reduce the complexity of trigonometric problems, making it easier to identify patterns and solutions. In the problem, the identity \( \sec^2 x = 1 + \tan^2 x \) serves as the starting point to manipulate and simplify the equation, leading to a clear path to solutions.