Problem 35 - Precalculus with Limits

Question

In Exercises 25-38, find all solutions of the equation in the interval \( [0, 2\pi) \). \( 2 \sec^2 x + \tan^2 x - 3 = 0 \)

Step-by-Step Solution

Answer

The solutions of the equation \(2 \sec^2 x + \tan^2 x - 3 = 0\) in the interval \([0, 2\pi)\) are \(x = \arctan(\sqrt{1/3}), \pi + \arctan(\sqrt{1/3}), \pi + \arctan(-\sqrt{1/3})\).

Step 1: Rewrite the equation

Replacing \(\sec^2 x\) with \(1 + \tan^2 x\) in the equation gives a quadratic equation in terms of \(\tan x\):\(2(1 + \tan^2 x) + \tan^2 x - 3 = 0\)Simplifying this we get,\(2\tan^2 x + 2 + \tan^2 x - 3 = 0\)Further simplifying, we have,\(3\tan^2 x - 1 = 0\)

Step 2: Solve the quadratic equation to find \(\tan x\)

To find the roots of the equation,\(3\tan^2 x - 1 = 0\),Rearrange the equation to get,\(3\tan^2 x = 1\),Divide by 3 on both sides,\(\tan^2 x = 1/3\),Taking square roots of both sides to get two values,\(\tan x = \pm \sqrt{1/3}\)

Step 3: Find x in the interval

The possible values of \(x\) can be obtained from the \(\tan x\) function for the interval \([0, 2\pi)\).\(x = \arctan(\sqrt{1/3}), \arctan(-\sqrt{1/3})\)The four solutions in the interval are \(x = \arctan(\sqrt{1/3}), \pi + \arctan(\sqrt{1/3}), \pi + \arctan(-\sqrt{1/3}), 2\pi + \arctan(-\sqrt{1/3})\) but since we cannot have 2\(\pi\) included, the three solutions are\(x = \arctan(\sqrt{1/3}), \pi + \arctan(\sqrt{1/3}), \pi + \arctan(-\sqrt{1/3})\)

Key Concepts

Quadratic Equations in Trigonometry

When studying trigonometry, students often encounter quadratic equations, which may not initially seem related to trigonometric functions. However, when trigonometric identities come into play, trigonometric expressions often transform into quadratic equations.

In the given exercise, we had to deal with a quadratic equation in the variable \tan x\. By utilizing the trigonometric identity \( \tan^2 x = \frac{1}{3} \) and solving for the variable \tan x\, we essentially transformed a trigonometry problem into a more familiar algebraic form—a typical quadratic equation. In this form, we could apply well-known techniques, like factoring or using the quadratic formula, to find the solutions for \tan x\.

Understanding how to handle trigonometric equations as quadratic equations allows students to apply their algebraic problem-solving skills in the realm of trigonometry. It bridges the gap between two areas of mathematics, providing a powerful tool for solving complex problems.

Inverse Trigonometric Functions

Inverse trigonometric functions are used to find the angle that corresponds to a given trigonometric value. They are the 'inverse operations' of the trigonometric functions and include \text{arcsin\(x\)}, \text{arccos\(x\)}, and \text{arctan\(x\)} among others.

In our problem, after finding the trigonometric values \tan x = \frac{1}{3}\, we used the \text{arctan\(x\)} function to find the angles that correspond to those \tan\ values within the given interval. The \text{arctan\(x\)} function outputs an angle whose tangent is \text{\(x\)}, which in this case were \text{arctan\(\sqrt{1/3\}\)} and \text{arctan\(-\sqrt{1/3\}\)}.

It's important to note that because the tangent function has a period of \text{\( \pi \)} radians, there can be multiple solutions to an equation involving \text{arctan\(x\)}. The student therefore must consider all possible angles that could yield the same tangent value within the specified interval.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable where both sides are defined. These identities allow complex expressions to be simplified and are essential when solving trigonometric equations.

In the solution to the exercise, the trigonometric identity \( sec^2x = 1 + \tan^2 x \) was used to rewrite the original equation into a quadratic form. This is a classic example of how identities are applied to simplify and solve problems. Knowing the fundamental identities, such as the Pythagorean identity (which was used here), helps to solve various types of trigonometric equations.

Students must familiarize themselves with a suite of core trigonometric identities, including reciprocal identities, Pythagorean identities, and angle sum and difference identities, all of which provide the foundation for advanced problem-solving in trigonometry.