Problem 40 - 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 values of x are \( \frac{\pi}{2} \) and \( \frac{5\pi}{2} \).

Step 1: Understand the problem

We need to find the values of \( x \) in the interval \([0, 4\pi]\) where \( \sin x = 1 \). This means we are looking for points on the sine graph where the sine value is maximum, i.e., 1.

Step 2: Recall the properties of the sine function

The sine function \( y = \sin x \) reaches the value of 1 at every \( \frac{\pi}{2} + 2\pi n \), where \( n \) is an integer. This is because sine has a period of \( 2\pi \).

Step 3: Determine the applicable range of x

We need to check which of these values fall within the interval \([0, 4\pi]\).

Step 4: Calculate the values of x

Starting from \( x = \frac{\pi}{2} \) and adding the period \( 2\pi \) until reaching \( 4\pi \):- \( x = \frac{\pi}{2} \)- \( x = \frac{\pi}{2} + 2\pi = \frac{5\pi}{2} \)- Adding another \( 2\pi \) yields \( x = \frac{9\pi}{2} \), but this is outside the interval.

Step 5: Verify the solutions

Check these values satisfy \( \sin x = 1 \), which they do, and ensure they fall within the specified interval. Both \( \frac{\pi}{2} \) and \( \frac{5\pi}{2} \) satisfy the requirements.

Key Concepts

Sine Function

The sine function, denoted as \( y = \sin x \), is a fundamental concept in trigonometry. It describes the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. However, it can also be understood in the context of the unit circle, which is essential to trigonometric analysis. The sine function outputs values that range from -1 to 1. It reaches its peak value of 1 at specific points, which occurs periodically along its wave-like graph. This periodic nature means it repeats every \( 2\pi \), a concept known as its period.

The sine function reaches the value of 1 at the angle \( \frac{\pi}{2} \) and repeats this maximum value at each interval of \( \frac{\pi}{2} + 2\pi n \). Here, \( n \) represents any integer, indicating the repetitive cycles of the function. This repetition is a key characteristic of trigonometric functions, making them valuable in modeling cyclical phenomena such as sound waves or tides.

Unit Circle

The unit circle is a powerful tool in understanding trigonometric functions. It is a circle with a radius of 1 centered at the origin of a coordinate system. The angle measurements on the unit circle can be taken in radians, where \( 2\pi \) radians equate to a full circle, and \( \pi \) radians equate to half a circle.

On the unit circle, any point \( (x, y) \) represents the cosine and sine of the angle \( \theta \):

• \( x = \cos \theta \)
• \( y = \sin \theta \)

This setup allows us to visually interpret the sine and cosine functions. When analyzing \( \sin x = 1 \), we refer to the topmost point on the unit circle, at an angle of \( \frac{\pi}{2} \) radians. Reaching this same top position after cycling around the circle means adding \( 2\pi \) radians each time, illustrating the periodic nature of the sine function.

Trigonometric Equations

Trigonometric equations involve trigonometric functions such as sine, cosine, and tangent. Solving these equations often involves finding all angle measures \( x \) that satisfy the equation within a specified interval. This may require using the inherent periodic properties of these functions.

Consider the example equation \( \sin x = 1 \). You must find all angles \( x \) for which this condition holds true within an interval, such as \([0, 4\pi]\).

• Start by identifying a foundational solution, like \( \frac{\pi}{2} \), where the sine function is first maximized within the basic cycle of \( [0, 2\pi] \).
• Account for all equivalent angles within the given interval by adding full cycles of \( 2\pi \), i.e., \( x = \frac{\pi}{2} + 2\pi n \).
• Verify which of these possible solutions fall within the required interval, ensuring their validity.

This methodical process emphasizes understanding and applying the periodic nature of trigonometric functions to solve equations comprehensively.