Problem 21 - Applied Calculus

Question

Find the equation of the tangent line to the graph of \(y=\sin x\) at \(x=\pi\). Graph the function and the tangent line on the same axes.

Step-by-Step Solution

Answer

The equation of the tangent line is \(y = -x + \pi\).

Step 1: Find the Point of Tangency

To find the point of tangency, we need to evaluate the function at the given point. The function is \(y = \sin x\), and the given \(x\) value is \(x = \pi\).\[y = \sin \pi = 0.\]Thus, the point of tangency is \((\pi, 0)\).

Step 2: Calculate the Derivative

The derivative of the function \(y = \sin x\) gives us the slope of the tangent line. The derivative of \(\sin x\) is \(\cos x\). Calculate the derivative at the point \(x = \pi\).\[y' = \cos x,\] \[y'(\pi) = \cos \pi = -1.\]Hence, the slope \(m\) of the tangent line at \(x = \pi\) is \(-1\).

Step 3: Use the Point-Slope Form to Write the Equation

With the slope from the derivative \(-1\) and the point \((\pi, 0)\), we use the point-slope form \(y - y_1 = m(x - x_1)\) to find the equation of the tangent line.\[y - 0 = -1(x - \pi).\] Simplify to get the equation: \[y = -x + \pi.\]

Step 4: Graph the Function and the Tangent Line

To visualize the function and the tangent line on the same axes, plot \(y = \sin x\) and the tangent line \(y = -x + \pi\). The sine function oscillates above and below the line \(y = 0\), while the tangent line at \(x = \pi\) will be a straight line with a negative slope passing through the point \((\pi, 0)\).

Key Concepts

Sine Function

The sine function is a fundamental trigonometric function that is known for its wave-like shape. It is defined by the equation \(y = \sin x\), where \(x\) is in radians. This function oscillates between -1 and 1, with key properties including:

• Periodic Nature: The sine function has a period of \(2\pi\), meaning it repeats its wave pattern every \(2\pi\) radians.
• Symmetry: It is an odd function, meaning that \(\sin(-x) = -\sin(x)\).
• Intercepts: It crosses the x-axis at integer multiples of \(\pi\).

Understanding the behavior of the sine function is essential for graphing and analyzing trigonometric-related problems.

Derivative

The derivative is a mathematical tool that measures how a function changes as its input changes. Specifically, it provides the slope of the tangent line to the function at any given point. For the sine function, the derivative is as follows:

• The derivative of \(y = \sin x\) is \(y' = \cos x\).
• At \(x = \pi\), the derivative of \(\sin x\) evaluates to \(\cos \pi\), which is \(-1\).

This tells us that the slope of the tangent line to the curve of \(y = \sin x\) at \(x = \pi\) is \(-1\). This negative slope indicates a downward trend immediate to the point \((\pi, 0)\).

Point-Slope Form

The point-slope form is a way to express the equation of a line. It is particularly useful when you know a point on the line and the slope. The formula is:

• \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line, and \(m\) is the slope.

In the context of our exercise, the point \((\pi, 0)\) is on the line, and the slope \(m\) is \(-1\). Plug these values into the point-slope form:

• \(y - 0 = -1(x - \pi)\)

Simplifying gives us the linear equation \(y = -x + \pi\), which represents the tangent line at the point where it touches the sine curve.

Graphing Functions

Graphing functions can provide valuable insight into the function's behavior and its interplay with other functions. When plotting \(y = \sin x\) and its tangent line \(y = -x + \pi\) on the same axes:

• The sine function will exhibit its familiar periodic wave pattern, oscillating between -1 and 1.
• The tangent line, \(y = -x + \pi\), is a straight line with a negative slope, intersecting the curve at \((\pi, 0)\).

This visualization makes it clear how the tangent line approximates the sine function at the point of tangency. Despite the curves crossing, they share the same slope at precisely \(x = \pi\), demonstrating why the tangent line is relevant in understanding local linearization.