Problem 48 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Equation

We start by analyzing the given equation: \( \sin x - \cos x = 0 \). Our goal is to find all \( x \) within the interval \([0, 2\pi)\) that make this equation true.

Step 2: Rewrite the Equation

To simplify the equation, let's rewrite it as \( \sin x = \cos x \). This indicates that the sine and cosine functions must be equal.

Step 3: Determine Standard Angles

The equation \( \sin x = \cos x \) is true for angles where the functions coincide, such as at a 45-degree angle, which is equivalent to \( \frac{\pi}{4} \) radians.

Step 4: Find General Solution

The general solution to \( \sin x = \cos x \) involves angles where \( x = \frac{\pi}{4} + n\pi \), because sine and cosine functions equal at this angle periodically every \( \pi \) radians.

Step 5: Find Specific Solutions in the Interval

Now, substitute values of \( n \) to find specific solutions within \([0, 2\pi)\). Start with \( n = 0 \): \( x = \frac{\pi}{4} \). For \( n = 1 \): \( x = \frac{5\pi}{4} \). Check that \( x = \frac{\pi}{4} \) and \( x = \frac{5\pi}{4} \) are both within the interval.

Key Concepts

Sine and Cosine Functions

Trigonometric functions like sine and cosine are fundamental in mathematics and describe relationships in right triangles and unit circles. The sine and cosine functions relate the angle of a right triangle to the ratio of its sides and give insight into circular motion around the unit circle. For any angle \( x \), the following properties are essential:

• The sine function, denoted as \( \sin x \), represents the y-coordinate of a point on the unit circle at angle \( x \).
• The cosine function, denoted as \( \cos x \), represents the x-coordinate of a point on the unit circle at angle \( x \).
• These functions are periodic with a period of \( 2\pi \), meaning they repeat their values every \( 2\pi \) radians.

Understanding the equality \( \sin x = \cos x \) involves recognizing angles where these functions have the same value, notably in specific quadrants on the unit circle. These key points help solve trigonometric equations efficiently.

Angle Measurement in Radians

Radians offer a natural and effective way to measure angles based on the radius of a circle. Unlike degrees, radians directly relate to the circle's geometry. Here’s why radians are valuable:

• A circle with a radius of 1 unit (a unit circle) has a circumference of \( 2\pi \) radians.
• One full circle is \( 2\pi \) radians, equivalent to 360 degrees.
• An angle's measurement in radians represents the length of the arc instead of arbitrary units like degrees.

For example, \( \frac{\pi}{4} \) radians is equivalent to 45 degrees. Using radians simplifies mathematical calculations, especially when dealing with trigonometric functions and calculus. When solving \( \sin x = \cos x \) for specific values, converting based on radians helps identify key angles like \( \frac{\pi}{4} \) and its periodic repetitions.

Solving for Specific Intervals

In trigonometry, solving equations often involves identifying values within specific intervals. This means finding all the solutions for an angle within a defined range. Here's how to approach this:

• First, recognize the equation's symmetries or periodic properties, especially for trigonometric equations.
• Determine the general solution, which represents all possible solutions across any interval.
• Apply the interval restrictions to narrow down to solutions within the specified range, such as \([0, 2\pi)\).

For instance, in solving \( \sin x = \cos x \), the general solution is \( x = \frac{\pi}{4} + n\pi \) due to the periodic nature of sine and cosine functions. By substituting different integers for \( n \), and then checking within \([0, 2\pi)\), we find specific solutions such as \( x = \frac{\pi}{4} \) and \( x = \frac{5\pi}{4} \). This technique allows you to capture the precise instances where the equality holds true within the targeted interval.