Question

Writing to Learn Graph \(y=\sin x\) and \(y=\cos x\) in the same viewing window. Which function could be the derivative of the other? Defend your answer in terms of the behavior of the graphs.

Short answer

The function \(y=\cos x\) could be the derivative of \(y=\sin x\), and the function \(y=-\sin x\) could be the derivative of \(y=\cos x\). This can be inferred by comparing the slope (derivative) of one function with the value of the other at corresponding points.

Step-by-step solution

Graphing Sine Function

First graph the function \(y=\sin x\) which starts at the origin (0,0), reaches a maximum at \(\frac{π}{2}\) , crosses the x-axis at π, reaches a minimum at \(\frac{3π}{2}\), and completes a full cycle at 2π.

Graphing Cosine Function

Next, graph the function \(y=\cos x\) which starts at its maximum (0,1), crosses the x-axis at \(\frac{π}{2}\), reaches a minimum at π, crosses the x-axis again at \(\frac{3π}{2}\) , and completes a full cycle at 2π.

Comparing Sine and Cosine Functions and Their Derivatives

The derivative of \(y=\sin x\) is \(y=\cos x\) and the derivative of \(y=\cos x\) is \(y=-\sin x\). The derivative represents the slope or rate of change of the function. Looking at the graphs, it is clear that when the \(y=\sin x\) graph is increasing, the \(y=\cos x\) graph is positive, implying that the \(y=\cos x\) graph could represent the derivative (slope) of the \(y=\sin x\) graph. On the other hand, when the \(y=\cos x\) graph is decreasing, the \(y=-\sin x\) graph is negative, which is consistent with the \(y=-\sin x\) graph being the derivative of the \(y=\cos x\) graph.

Key concepts

Derivatives of Trigonometric Functions

Understanding the derivatives of trigonometric functions is crucial for students delving into calculus. When we talk about the derivative of a function, we refer to the instant rate of change of the function at any given point. In other words, it's the function's slope, or the 'speed' at which the y-value is changing as x increases.

For the sine function, given by the equation \(y = \text{sin}(x)\), its derivative is the cosine function, \(y' = \text{cos}(x)\). Conversely, the cosine function, \(y = \text{cos}(x)\), has a derivative that is the negative sine function, \(y' = -\text{sin}(x)\). This means that if you were to take a tiny step along the \(x\)-axis and look at how fast the \(y\)-value of the sine function is changing at that point, you'd find that it's changing at the rate given by the value of the cosine function at that same point.

• Derivative of \(\text{sin}(x)\) is \(\text{cos}(x)\)
• Derivative of \(\text{cos}(x)\) is \(-\text{sin}(x)\)

These relationships are critical in many areas of mathematics and physics, where the understanding of motion and waves often relies on trigonometric functions and their derivatives.

Comparing Sine and Cosine Graphs

The sine and cosine functions are foundational in trigonometry and have graphs that showcase periodic behavior. These functions are similar in many ways, yet there are key differences that are important to understand.

The graph of \(y = \text{sin}(x)\) starts at the origin (0,0), indicating that the sine of 0 is 0. As \(x\) increases, the graph rises to a peak, declines back to zero, continues to a trough, and then rises back to zero, completing one full cycle every \(2\text{\pi}\) radians (360 degrees).

The graph of \(y = \text{cos}(x)\) starts at its maximum value of 1 when \(x = 0\). This is because the cosine of 0 is 1. The graph follows a similar pattern to the sine graph but shifted to the left by \(\frac{\text{\pi}}{2}\) radians. This shift means that the peaks, troughs, and zeros of the cosine function occur a quarter cycle before those of the sine function.

Key Points of Comparison:

• Sine starts at zero; cosine starts at a maximum.
• The maximum of sine is at \(\frac{\text{\pi}}{2}\); the maximum of cosine is at 0.
• Both functions repeat every \(2\text{\pi}\) radians.
• Sine is shifted \(\frac{\text{\pi}}{2}\) radians to the right of cosine.

By visualizing these graphs, students can better grasp the periodic nature of these functions and notice that the derivative of sine shows the slope of the sine curve at every point, one of the many interesting overlaps between algebra and calculus.

Calculus in Trigonometry

When we integrate calculus with trigonometry, we're able to solve more complex problems involving rates of change and areas under curves—topics that are central to calculus. Calculus comes into play in trigonometry primarily through the concepts of differentiation and integration applied to trigonometric functions.

As seen with the sine and cosine functions, understanding their derivatives\—\text{cos}(x)\ and \(-\text{sin}(x)\) respectively\—is just the tip of the iceberg. Integration is another calculus concept that's equally important. It involves finding the area under the curve of a function. For instance, integrating the \(\text{cos}(x)\) function gives you the \(\text{sin}(x)\) function, and integrating the \(-\text{sin}(x)\) function gives you the \(-\text{cos}(x)\) function.

Both differentiation and integration transform one trigonometric function into another, revealing a symmetry that is both elegant and practically significant. These techniques are not just purely academic exercises; they have applications in physics, engineering, and other sciences, where trigonometric functions model phenomena ranging from sound waves to the behavior of alternating currents.

Applications of Trigonometric Calculus:

• Finding slopes and areas related to trigonometric functions
• Analyzing the behavior of waves and oscillations
• Optimizing functions to solve real-world problems

Thus, calculus does not operate in isolation; it is deeply intertwined with trigonometry, providing tools to unravel more about the world around us and the patterns that describe it.