Problem 76 - Precalculus

Question

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

Step-by-Step Solution

Answer

Graph \(y = \cos x - \sin x\) over \(0 \leq x \leq 2\pi\) using key points: (0,1), \(\left(\frac{\pi}{2}, -1\right)\), \(\left(\pi, -1\right)\), \(\left(\frac{3\pi}{2}, 1\right)\), (\2\pi,1\).

Step 1: Understand the Individual Functions

We have two individual trigonometric functions: \(y_1 = \cos x\) and \(y2 = \sin x\). The task is to find the sum \(y = \cos x - \sin x\) and graph it over the interval \(0 \leq x \leq 2\pi\).

Step 2: Express the Summed Function

The summed function is given by: \(y = \cos x - \sin x\). This function is derived by adding the ordinates (y-values) of the individual functions: the ordinate of \(\cos x\) and subtracting the ordinate of \(\sin x\).

Step 3: Determine Key Points on the Interval

Evaluate \(y = \cos x - \sin x\) at key intervals, particularly at \(x = 0\), \(x = \frac{\pi}{2}\), \(x = \pi\), \(x = \frac{3\pi}{2}\), and \(x = 2\pi\):- \(y(0) = \cos(0) - \sin(0) = 1 - 0 = 1\)- \(y\left(\frac{\pi}{2}\right) = \cos\left(\frac{\pi}{2}\right) - \sin\left(\frac{\pi}{2}\right) = 0 - 1 = -1\)- \(y(\pi) = \cos(\pi) - \sin(\pi) = -1 - 0 = -1\)- \(y\left(\frac{3\pi}{2}\right) = \cos\left(\frac{3\pi}{2}\right) - \sin\left(\frac{3\pi}{2}\right) = 0 - (-1) = 1\)- \(y(2\pi) = \cos(2\pi) - \sin(2\pi) = 1 - 0 = 1\).

Step 4: Sketch the Graph of the Summed Function

Use the evaluated points to sketch the graph. Connect these points with a smooth curve. Note that the function will oscillate between 1 and -1, taking a periodic shape similar to the individual sine and cosine waves but phase shifted due to subtraction.

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides in a right-angled triangle. These functions are fundamental in trigonometry and are used to model periodic phenomena such as sound and light waves. Common trigonometric functions include sine (\(\sin \)), cosine (\(\cos \)), and tangent (\(\tan \)). Each of these functions represents a specific relationship between the angles and sides of a triangle.

• Sine Function: Denoted as \(\sin(x) \), it represents the y-coordinate of a point on the unit circle associated with an angle x.
• Cosine Function: Written as \(\cos(x) \), it denotes the x-coordinate of that point on the unit circle.
• Tangent Function: The ratio of sine to cosine, \(\tan(x) = \frac{\sin(x)}{\cos(x)} \).

Trigonometric functions are periodic, meaning they repeat their values in regular intervals. This property makes them particularly useful for describing cycles, such as the oscillation of waves.

Sine and Cosine

Sine and cosine are the foundational trigonometric functions, sharing a unique relationship because of their origin from the unit circle. They both have a period of \(2\pi\), which means they repeat every \(360^{\circ}\).
Important Characteristics of Sine and Cosine:

• Both functions oscillate between -1 and 1.
• They are out of phase by \(\frac{\pi}{2} \) radians; this means that when one is at a maximum, the other is at zero.
• Even and Odd Functions: Cosine is an even function, satisfying \(\cos(-x) = \cos(x) \), while sine is an odd function, satisfying \(\sin(-x) = -\sin(x) \).

Sine and cosine are often used together to form other functions, such as in the equation \(y = \cos(x) - \sin(x) \), where each function contributes to the overall behavior of the wave pattern.

Function Summation

Function summation involves the addition or subtraction of two or more functions. In trigonometry, combining functions like sine and cosine can yield interesting results. For example, the expression \(y = \cos(x) - \sin(x) \) is a summation of two fundamental trigonometric functions except one is subtracted.
When adding or subtracting these functions:

• Different Amplitudes and Phases: The resulting function may have altered amplitude and phase compared to the original functions.
• Shifts and Interferences: The wave pattern changes, demonstrating the interference of waves; some cycles may amplify or reduce in magnitude.

Function summation can be visualized easily by graphing, allowing one to see how peaks and troughs of the individual functions affect the resultant wave.

Graph Sketching

Graph sketching is a powerful technique for understanding and visualizing mathematical functions, especially trigonometric ones. By graphing the summed function \(y = \cos(x) - \sin(x) \), we can observe the new wave form in relation to its components.

• Key Points Identification: Evaluate the function at essential points within one period, such as \(x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, \text{and} 2\pi \).
• Smooth Connective Curves: Plot the points accurately and connect them to form a continuous wave.
• Pattern Recognition: Identify the periodicity and range, observing how the function behaves especially at maximum and minimum points.

By sketching, we can visually interpret how the subtraction of sine from cosine affects the resultant pattern, revealing specifics like phase shifts, node points, and symmetry, which are harder to grasp algebraically.