Problem 48 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 39-62: Find the solutions of the equation that are in the interval \([0,2 \pi\) ). $$ \sin x-\cos x=0 $$

Step-by-Step Solution

Answer

Solutions are \( x = \frac{\pi}{4} \) and \( x = \frac{5\pi}{4} \).

Step 1: Set the equation for trigonometric functions

Start solving the equation by setting the equation \[ \sin x - \cos x = 0 \]in terms of a trigonometric identity.

Step 2: Rearrange the equation

Rearrange the equation to isolate one trigonometric function. We can write it as:\[ \sin x = \cos x \]

Step 3: Transform to tangent function

Next, divide both sides by \(\cos x\):\[ \frac{\sin x}{\cos x} = \frac{\cos x}{\cos x} \]which reduces to:\[ \tan x = 1 \]

Step 4: Solve for x using the tangent function

The equation \( \tan x = 1 \) is true when the angle \( x \) corresponds to which the tangent of the angle is 1. That occurs at\[ x = \frac{\pi}{4} + n\pi \]where \( n \) is an integer.

Step 5: Find solutions in the interval [0, 2π)

List the solutions \( x = \frac{\pi}{4} + n\pi \) in the interval \([0, 2\pi)\):1. For \( n = 0 \), \( x = \frac{\pi}{4} \).2. For \( n = 1 \), \( x = \frac{\pi}{4} + \pi = \frac{5\pi}{4} \).These are the only solutions within the given interval.

Key Concepts

Solving Equations

Solving equations involves finding values that satisfy the mathematical statement presented. In this exercise, we encounter a trigonometric equation: \( \sin x - \cos x = 0 \). The goal is to find values of \( x \) that make the equation true. To achieve this, the equation should be manipulated into a form that reveals these solutions more clearly.

• Rewriting the equation: Start by expressing the equation in a simpler form using algebraic manipulations. For this equation, rewrite it to \( \sin x = \cos x \).
• Utilizing known functions: Transform the equation to involve a single trigonometric function, such as tangent, which can simplify solving.
• Identifying solutions: Use inversions of trigonometric functions or unit circle knowledge to pinpoint exact solutions.

Moving from a complex starting point to a single function or expression is key in solving equations effectively, particularly when trigonometric identities are involved.

Trigonometric Identities

Trigonometric identities are formulas involving trigonometric functions that hold true for all values of the variables. They act as powerful tools to simplify and solve trigonometric equations. In this exercise, the primary trigonometric identity used is the relation between sine and cosine to tangent.

• Basic identity: Because \( \tan x = \frac{\sin x}{\cos x} \), we can transform equations to make use of this relationship.
• Unit circle relationships: Remember that some specific angles have well-known trigonometric values, such as when \( \tan x = 1 \), corresponding to \( x = \frac{\pi}{4} + n\pi \) where \( n \) is an integer.

In trigonometry, being familiar with identities allows for greater flexibility and creativity in manipulating and solving equations. They bridge connections between different functions, streamlining the process of finding solutions.

Tangent Function

The tangent function is a crucial trigonometric function defined as the ratio of sine to cosine: \( \tan x = \frac{\sin x}{\cos x} \). It is periodic and has a period of \( \pi \), meaning it repeats its pattern every \( \pi \, \, \text{radians} \). This periodicity is important when determining solutions within a specified interval.

• Understanding \( \tan x \): Recognize that the function can have values ranging from negative infinity to positive infinity, specifically when the sine and cosine functions intersect.
• Finding angles: Solutions for equations like \( \tan x = 1 \) occur at specific angles, found using both the unit circle and the periodic property of the tangent function.
• Interval constraints: Solutions such as \( x = \frac{\pi}{4} + n\pi \) are evaluated within a defined interval like \([0, 2\pi)\), to ensure only appropriate solutions are considered for the given problem context.

Mastering the properties of the tangent function aids immensely in navigating through trigonometric equations, ensuring accurate and efficient solutions.