Problem 91 - Precalculus

Question

In Exercises \(67-94,\) add the ordinates of the individual functions to graph each summed function on the indicated interval. $$y=2 \sin \left(\frac{x}{2}\right)-\cos (2 x), 0 \leq x \leq 4 \pi$$

Step-by-Step Solution

Answer

Graph the function over \([0, 4\pi]\) using the calculated key points: (0, -1), (\pi, 1), (2\pi, -1), (3\pi, -3), (4\pi, -1).

Step 1: Understand the Problem

We are required to graph the summed function over the interval \([0, 4\pi]\). The function is given as \(y = 2 \sin \left(\frac{x}{2}\right) - \cos(2x)\). We will use specific points within this interval to calculate the value of \(y\) by adding the ordinates of the individual terms \(2 \sin \left(\frac{x}{2}\right)\) and \(-\cos(2x)\).

Step 2: Evaluate Function at Key Points

Select key points within the interval \(0 \leq x \leq 4\pi\), such as \(x = 0, \pi, 2\pi, 3\pi, 4\pi\), and evaluate the function at these points.

Step 3: Calculate y Values at Key Points

Evaluate the function at the key points:- At \(x = 0\), \(y = 2 \sin(0) - \cos(0) = 0 - 1 = -1\).- At \(x = \pi\), \(y = 2 \sin\left(\frac{\pi}{2}\right) - \cos(2\pi) = 2 - 1 = 1\).- At \(x = 2\pi\), \(y = 2 \sin(\pi) - \cos(4\pi) = 0 - 1 = -1\).- At \(x = 3\pi\), \(y = 2 \sin\left(\frac{3\pi}{2}\right) - \cos(6\pi) = - 2 - 1 = -3\).- At \(x = 4\pi\), \(y = 2 \sin(2\pi) - \cos(8\pi) = 0 - 1 = -1\).

Step 4: Plot the Calculated Points

Use the calculated \(y\)-values at each key point \((x,y) = (0, -1), (\pi, 1), (2\pi, -1), (3\pi, -3), (4\pi, -1)\) as representative values.Plot these points on a graph with \(x\)-axis ranging from \(0\) to \(4\pi\) and connect them to understand the overall shape of the graph.

Step 5: Sketch the Graph

With the plotted points, draw smooth curves between points, taking into account the oscillating behavior of the trigonometric functions. This smooth plot will represent the summed function \(y = 2 \sin \left(\frac{x}{2}\right) - \cos(2x)\) across the interval \([0, 4\pi]\).

Key Concepts

Trigonometric Functions

Trigonometric functions are a critical part of mathematical analysis. At their core, they relate the angles of a triangle to the lengths of its sides. These functions, like sine and cosine, help us understand oscillating systems in mathematics and our natural world. A few key features of trigonometric functions include their periodicity, which means they repeat values in regular intervals. This makes them useful for modeling cycles, such as sound waves.

• Function Types: The primary trigonometric functions include sine (\( ext{sin} \)) and cosine (\( ext{cos} \)). Both are periodic, with a fundamental period of \(2\pi\).
• Ratios: Sine represents the ratio of the opposite side to the hypotenuse in a right triangle, and cosine represents the ratio of the adjacent side to the hypotenuse.
• Applications: Trigonometric functions apply in numerous fields including engineering, physics, and even economics.

For this exercise, the functions we are dealing with are transformations and combinations of the sine and cosine functions, representing more complex patterns.

Sine and Cosine Functions

Sine and cosine functions are foundational components of trigonometry. They serve as building blocks for complex equations and are defined by their distinct wave-like shapes.

• Sine Function: The sine function, \( y = ext{sin}(x) \), oscillates between \(-1\) and \(1\). It crosses the x-axis at multiples of \( \pi \), starting from 0. The function has a period of \(2\pi\).
• Cosine Function: The cosine function, \( y = ext{cos}(x) \), also ranges between \(-1\) and \(1\), but its waveform starts at \(y=1\) when \(x=0\). With the same period as sine, it precisely follows the sine function but is shifted.

In the exercise, the function \( y = 2 ext{sin} \left(\frac{x}{2}\right) - ext{cos}(2x) \) involves scaling and compressing these basic sine and cosine functions, adding layers of complexity to the graph.

Graphing Techniques

When graphing trigonometric functions, several approaches help in creating accurate graphs. This is particularly relevant for functions like \( y = 2 ext{sin} \left(\frac{x}{2}\right) - ext{cos}(2x) \), which involve transformed trigonometric components.

• Key Points: Choosing specific values of \(x\) where the trigonometric function's behavior is predictable, like \(0\), \(\pi\), and \(2\pi\), is crucial. Evaluating the function at these points gives the heights (ordinates) directly related to sine and cosine values.
• Transformations: Recognizing transformations such as stretching, compressing, and translating helps in sketching complex graphs. These transformations are induced by factors like multiplying a function or changing the input variable.
• Smooth Curves: Once key points and behaviors are plotted, connecting them with smooth curves is vital to physically represent continuity, given that trigonometric functions are continuous and naturally oscillating.

Proper application of these techniques allows for visualizing complicated trigonometric expressions and understanding their dynamic behavior over specific intervals.