Problem 38 - Precalculus with Limits

Question

In Exercises 25-38, find all solutions of the equation in the interval \( [0, 2\pi) \). \( \sin x - 2 = \cos x - 2 \)

Step-by-Step Solution

Answer

The solutions to the equation \( \sin x = \cos x \) in the interval [0, 2π) are \( x = \frac{\pi}{4}, \frac{5\pi}{4} \).

Step 1: Simplification of the Equation

First, it is important to simplify the given equation. This can be done by removing similar terms on either side. \So, let’s start simplifying \\( \sin x - 2 = \cos x - 2 \) \Upon simplification, we get \\( \sin x = \cos x \)

Step 2: Solve the Simplified Equation

The trigonometric equation \( \sin x = \cos x \) holds true for two conditions in the intervall [0, 2π): \First, when \( x = \frac{\pi}{4} \) \Second, \( x = \frac{5\pi}{4} \) \as these are the points where the sine and cosine functions coincide.

Step 3: Formulate the Solution Set

Having identified these points, we can generate the solution set for this exercise. So, the solution set of the equation \( \sin x = \cos x \) in the interval [0, 2π) is \( x = \frac{\pi}{4}, \frac{5\pi}{4} \).

Key Concepts

Sine and Cosine Functions

Understanding the sine and cosine functions is vital when it comes to solving trigonometric equations. The sine function (\( \text{sin} \theta \) for an angle \( \theta \) represents the y-coordinate of the point on the unit circle that corresponds to that angle. Similarly, the cosine function (\( \text{cos} \theta \) represents the x-coordinate for the same point on the unit circle.

It's important to remember that both of these functions are periodic, meaning they repeat their values in a regular cycle. The sine and cosine functions have a period of \( 2\text{π} \) which means after this interval, their values start repeating. Knowing the properties and graphs of these functions allows us to solve equations where they appear.

Trigonometric Solutions Interval

In trigonometry, the solutions to an equation can potentially be infinite because trigonometric functions are periodic. However, we often look for solutions within a specific interval. For many problems, including the one featured here, we are interested in solutions within the interval \( [0, 2\text{π}) \).

This interval means we are seeking the solutions that fall between 0 and just under \( 2\text{π} \), hence covering one complete cycle of the sine and cosine functions. Any solution outside of this interval is a repetition of a solution within it. By limiting the interval, we simplify the problem to a manageable number of solutions that can be easily visualized and understood.

Simplification of Equations

Simplification is essential in solving trigonometric equations. It often involves combining like terms, using algebraic manipulation, and applying arithmetic operations to both sides of the equation to isolate the trigonometric function you're trying to solve for.

In our exercise, for instance, subtracting 2 from both sides didn't change the equation much because it was canceled out on both sides. Simplification helped us reduce the equation to \( \text{sin} x = \text{cos} x \) which is easier to solve. Always aim for the simplest form of the equation, as it typically reveals a clearer path toward the solution.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved functions' variables. They are useful in simplifying expressions and solving trigonometric equations. One basic identity that is relevant in many situations is the Pythagorean identity, which states that for any angle \( \theta \) we have \( \text{sin}^2 \theta + \text{cos}^2 \theta = 1 \).

Knowledge of various identities allows you to transform the original equation into a more familiar form. For the exercise above, understanding that \( \text{sin} x = \text{cos} x \) implies that both sides are equivalent at certain angles - in the case of our problem, when \( x \) is \( \frac{\text{π}}{4} \) or \( \frac{5\text{π}}{4} \) because these are the angles where the values of sine and cosine are equal, and thus, they 'coincide'.