Problem 36 - Contemporary Precalculus

Question

Use the graphs of the trigonometric functions to determine the number of solutions of the equation between 0 and \(2 \pi\) \(\sin t=k,\) where \(k\) is a nonzero constant such that \(-1

Step-by-Step Solution

Answer

Answer: The equation \(\sin t=k\) has two solutions within the interval 0 to \(2\pi\) when -1 < k < 1.

Step 1: Understanding the sine function graph

The sine function graph is a periodic wave that oscillates between -1 and 1 on the y-axis. The period of the sine function is \(2\pi\), which means it repeats its pattern every \(2\pi\). Between 0 and \(2\pi\), the sine function starts at 0, goes to 1, then back to 0, goes to -1, and again goes back to 0.

Step 2: Identify the range of k

Given, \(-1< k <1\), which means the k value is real and lies within the range of -1 and 1. This range covers all possible values of the sine function.

Step 3: Determine the points of intersection between the sine graph and k

The points of intersection between the sine graph and k will represent the solutions of the equation \(\sin t=k\). As the sine function oscillates between -1 and 1 within the interval of \(0\) to \(2\pi\), it will have two points of intersection with the constant k.

Step 4: Conclude the number of solutions

Between 0 and \(2\pi\), the sine function will intersect the line \(y=k\) in two points. Therefore, the equation \(\sin t=k\) has two solutions within the given interval.

Key Concepts

Sine Function Graph

The sine function is one of the fundamental concepts in trigonometry, and understanding its graph is crucial for solving trigonometric equations. The sine graph represents a smooth, continuous wave that oscillates above and below the horizontal axis. This axis usually represents the angle in radians, while the vertical axis represents the sine value of that angle.

Specific characteristics of the sine graph include its amplitude, period, and phase shift. The amplitude is the height of the wave's peak, which for the basic sine function is 1, meaning it ranges from -1 to 1 on the y-axis. The period, an essential feature, is the length of one full cycle of the wave. For the sine function, the period is always \(2\pi\radians\), indicating the pattern repeats after every \(2\pi\) radians.

Considering a line drawn at a certain y-value \(k\) where \(k\) is between -1 and 1, the line will intersect the sine wave exactly twice within one period. These intersections are the visual representation of the solutions to the trigonometric equation \({\sin t = k}\) within the specified interval of 0 to \(2\pi\). Understanding this graph provides a clear insight into the periodic nature of sine and helps to visually solve equations based on it.

Periodic Functions

Periodic functions are those functions that repeat their values in regular intervals or periods. Trigonometric functions like sine, cosine, and tangent are classic examples of periodic functions. As seen in the sine graph, these functions exhibit repeating patterns after a certain length along the x-axis, which is known as the period.

The importance of recognizing a function as periodic lies in the predictability of its behavior. For these functions, knowing one period allows us to understand the entire function. This predictability greatly simplifies the task of solving equations, especially in trigonometry, where angles can have infinite repetitions.

When dealing with trigonometric equations, it is often useful to limit the analysis to one period. This approach allows us to find all unique solutions within that period, which can then be extrapolated to additional periods if necessary using the property \({\sin (t + 2n\pi) = \sin t}\), where \(n\) is an integer. This demonstrates how periodicity can be employed to both find and understand the infinite set of solutions to trigonometric equations.

Graphical Solution of Equations

A graphical approach to solving equations involves visually identifying where two graphs intersect, which is useful for understanding solutions of trigonometric equations like \(\sin t = k\). By plotting both the function graph \(y = \sin t\) and the line \(y = k\) on the same set of axes, the points of intersection represent the solutions to the equation.

To effectively use a graphical solution, one must be familiar with the concept of reading and interpreting graphs. The x-coordinates of the intersection points correspond to the values of \(t\) that satisfy the equation. Utilizing this method provides an intuitive and immediate visual comprehension of the solutions, making it easier for students to grasp the concept without having to rely solely on analytical methods.

It's also essential to consider the domain of the function to determine the relevant intersections. For example, if we are limited to finding solutions between 0 and \(2\pi\), we only consider the intersections within this range. This technique emphasizes the importance of the visual aspects of mathematics and how they complement traditional algebraic approaches in finding solutions to equations.