Problem 39 - Precalculus: Functions and Graphs

Question

Refer to the graph of \(y=\sin x\) or \(y=\cos x\) to find the exact values of \(x\) in the interval \([0,4 \pi]\) that satisfy the equation. $$\sin x=-1$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{3\pi}{2}\) and \(x = \frac{7\pi}{2}\).

Step 1: Understanding the Sine Function

The sine function \( y = \sin x \) is a periodic function with a period of \( 2\pi \). It oscillates between -1 and 1.

Step 2: Identifying the Given Value

We need to find where the sine function equals \(-1\) within the interval \([0, 4\pi]\). The sine function reaches -1 only at specific points within one period.

Step 3: Finding the First Occurrence

Within the interval \([0, 2\pi]\), \(\sin x = -1\) at \(x = \frac{3\pi}{2}\). This is because \(\sin x\) attains its minimum value of -1 at \(\frac{3\pi}{2}\).

Step 4: Considering the Periodicity of Sine

Since the sine function is periodic with period \( 2\pi \), every \(2\pi\) we reach the same values again. Therefore, we add or subtract \(2\pi\) to the first occurrence to find others within the interval \([0, 4\pi]\).

Step 5: Calculating Other Solutions

Starting from \(\frac{3\pi}{2}\), add \(2\pi\) to find another solution in the given interval. This gives us another solution at \(x = \frac{3\pi}{2} + 2\pi = \frac{7\pi}{2}\).

Step 6: Listing All Solutions

The solutions for \(\sin x = -1\) in the interval \([0, 4\pi]\) are \(x = \frac{3\pi}{2}\) and \(x = \frac{7\pi}{2}\).

Key Concepts

Sine Function

The sine function, denoted by \( y = \sin x \), is a fundamental trigonometric function used to model periodic phenomena like sound waves or tides. It describes the projection of a rotating vector on a unit circle:

• Its domain is all real numbers, since you can input any angle into the sine function.
• The range of the sine function, however, is limited to values between -1 and 1.
• These values account for the maximum and minimum heights of the wave-like graph of the function.

The sine function's graph is a smooth, continuous wave, known as a sinusoidal wave, that oscillates above and below the x-axis.

Periodicity

The concept of periodicity is essential in understanding trigonometric functions like the sine function. A function is periodic if it repeats its values at regular intervals. For the sine function:

• The period is \(2\pi\), meaning the function's graph repeats its shape every \(2\pi\)units.
• This property implies that \( \sin(x + 2\pi) = \sin x \) for any angle \(x\).

This periodic nature helps in identifying multiple solutions of trigonometric equations over large intervals. Knowing that the sine function repeats every \(2\pi\), we can derive new solutions by adding or subtracting multiples of \(2\pi\) from known solutions.

Interval Solutions

Finding solutions to trigonometric equations, like \( \sin x = -1 \), often involves determining the x values within a specific interval. In this problem, the task is to find such solutions in the interval \([0, 4\pi]\). Here's how to do it:

• Identify all points within one period, \([0, 2\pi]\), where the sine function reaches the target value.
• The sine function reaches a value of -1 at \( x = \frac{3\pi}{2} \).
• To find solutions beyond the first period, add the period (\(2\pi\)) to the initial solution, fitting within the given interval.
• This results in another solution point at \( x = \frac{7\pi}{2} \) since \( \frac{3\pi}{2} + 2\pi = \frac{7\pi}{2} \).

The endpoint constraint requires all solutions to lie within the \([0, 4\pi]\) boundary.

Graphical Analysis

Graphical analysis is a powerful method to visualize and solve equations involving trigonometric functions. For \( y = \sin x \):

• The graph of the sine function helps you locate values of \(x\) where \( \sin x = -1 \).
• The minimum points on the sine wave, where the curve touches its lowest position, indicate where \( y = -1 \).
• Visual inspection quickly reveals that this occurs at \(x = \frac{3\pi}{2} \) in the first cycle.

Beyond simple observations, the graph allows students to explore symmetry, translations, and transformations of trigonometric graphs, offering deeper insights into periodic behavior.