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: Use Trigonometric Identities

Recall the Pythagorean identity: \[ \sec^2 x = 1 + \tan^2 x \] Substitute this into the equation:\[ 1 + \tan^2 x = \sqrt{3} \tan x + 1 \] This simplifies to:\[ \tan^2 x = \sqrt{3} \tan x \]

Step 2: Rearrange into Quadratic Form

Rearrange the equation to make it easier to solve:\[ \tan^2 x - \sqrt{3} \tan x = 0 \]This can be factored as:\[ \tan x (\tan x - \sqrt{3}) = 0 \]

Step 3: Solve each Factor

Set each factor equal to zero and solve for \(x\):1. \( \tan x = 0 \)2. \( \tan x = \sqrt{3} \)

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

\(\tan x = 0\) within the interval \([0, \pi)\) gives:\[ x = 0 \text{ or } x = \pi \]Since we need solutions in the interval \([0, \pi)\), we have:\[ x = 0 \]

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

The angles where \( \tan x = \sqrt{3} \) correspond to:\[ x = \frac{\pi}{3} \] in the interval \([0, \pi)\).

Step 6: Combine the Solutions

The solutions to the equation on the interval \([0, \pi)\) are:\[ x = 0 \] and\[ x = \frac{\pi}{3} \]

Key Concepts

Trigonometric Identities

Trigonometric Identities are essential tools in solving trigonometric equations. They help us rewrite expressions in ways that allow us to solve equations more easily. These identities, like the Pythagorean identity, relate the different trigonometric functions to one another. Some of the most common ones include:

• Reciprocal Identities: Used for relating secant, cosecant, and cotangent to sine, cosine, and tangent.
• Pythagorean Identities: These are based on the Pythagorean theorem and relate square functions such as sine, cosine, and tangent.
• Angle Sum and Difference Identities: Help in expanding trigonometric functions at angles expressed as sums or differences.

Understanding these identities and knowing how to apply them simplifies complex trigonometric expressions. The equation in our exercise utilized the Pythagorean identity to transform and simplify our problem for better manageability.

Pythagorean Identity

The Pythagorean Identity is a cornerstone of trigonometry. It arises from the Pythagorean theorem and relates the squares of the sine and cosine functions. The identity we used in our exercise is:\[\sec^2 x = 1 + \tan^2 x\]This identity links secant, a reciprocal of cosine, with tangent, and makes it possible to substitute and simplify the original equation. Applying these identities, we simplified the equation:\[\sec^2 x = \sqrt{3} \tan x + 1\]by transforming it using:\[1 + \tan^2 x = \sqrt{3} \tan x + 1.\]When attempting to resolve trigonometric equations, recognizing these relationships can substantially ease the process of finding solutions. This approach helps focus on the core relationships between the functions rather than becoming overwhelmed by the complexity of the terms.

Solving Quadratic Equations

Solving Quadratic Equations is a vital skill, particularly when these equations appear within trigonometry problems. After using the Pythagorean identity, our equation takes on a quadratic structure. This is common when dealing with trigonometric equations, as it allows us to employ algebraic techniques. The equation:\[\tan^2 x - \sqrt{3} \tan x = 0\]resembles the standard quadratic form \(ax^2 + bx + c = 0\). To solve this, you can use various methods such as factoring, completing the square, or using the quadratic formula. Here, we factor the equation:\[\tan x(\tan x - \sqrt{3}) = 0\]This results in two separate equations:

• \(\tan x = 0\)
• \(\tan x = \sqrt{3}\)

By solving these, we identify the solutions for \(x\) within the specified interval. Quadratic equations often simplify trigonometric problems by breaking them into manageable parts.

Interval Solutions

Interval Solutions refer to finding answers that fit within specified boundaries. In real-world applications and many trigonometric problems, solutions are constrained to a particular interval. In our exercise, the interval is \([0, \pi)\), which means we look for solutions where \(0 \leq x < \pi\). When solving \(\tan x = 0\), we find solutions like:

• \(x = 0\)
• \(x = \pi\)

However, the interval \([0, \pi)\) excludes \(\pi\), leaving us with:

• \(x = 0\)

Similarly, for \(\tan x = \sqrt{3}\), we identify the solution as:

• \(x = \frac{\pi}{3}\)

By ensuring solutions lie within the interval, we refine our results to meet the exercise's requirements. This approach highlights the need to always focus on interval constraints when finalizing solutions.