Problem 31 - Technical Mathematics with Calculus

Question

Tangents Find the slope of the tangent to four significant digits at the given value of \(x .\) $$y=\sin x \quad \text { at } x=2 \text { rad }$$

Step-by-Step Solution

Answer

The slope of the tangent to the curve y = sin(x) at x = 2 radians is approximately -0.4161.

Step 1: Find the derivative of the function

To find the slope of the tangent, first find the derivative of the function with respect to x. The derivative of the function y = sin(x) with respect to x is y' = cos(x).

Step 2: Evaluate the derivative at the given point

Now, evaluate the derivative (y' = cos(x)) at the given value of x, which is x = 2 radians. This gives us the slope of the tangent at that point: slope = cos(2).

Step 3: Calculate the numerical value

Calculate the numerical value of cos(2) using a calculator or by referring to a trigonometric table, ensuring to round to four significant digits.

Key Concepts

Derivative of Sine Function

Understanding the derivative of the sine function is crucial for various applications in calculus. The sine function, denoted as \(\sin x\), is one of the primary trigonometric functions, and it describes the ratio of the opposite side to the hypotenuse in a right-angled triangle. When taking the derivative of \(\sin x\) with respect to \(x\), we use the fundamental rules of differentiation to arrive at its derivative, which is \(\cos x\). Mathematically, this is expressed as \(\frac{d}{dx}\sin x = \cos x\).

Why is this important? When calculating the slope of the tangent line to a curve at a particular point, the derivative gives us exactly that. The slope of the tangent line to the graph of \(y=\sin x\) at any point \(x\) is the value of \(\cos x\) at that specific point. This is because the cosine function represents the rate of change of the sine function, and thus, the steepness of the tangent at any given point.

Evaluating Derivatives

Evaluating derivatives is a fundamental process in calculus that involves computing the value of a derivative at a particular point. Once we have the derivative function, as in the example of the sine function above, we want to find its value at a specific \(x\). This is effectively what we do to find the slope of the tangent line.

To evaluate the derivative \(y' = \cos x\) at \(x = 2\) radians, we simply substitute \(x\) with 2 in the cosine function, resulting in \(\cos(2)\). The value obtained from this calculation provides the exact slope of the tangent line to the curve \(y = \sin x\) at \(x = 2\) radians. It is important to be familiar with the unit circle, the calculator, or trigonometric tables to find the numerical approximation, and to round it according to the required level of precision.

Trigonometric Functions

Trigonometric functions are the cornerstone of trigonometry and are extensively used in evaluating the properties of angles and their relationships to the sides of triangles. Besides the sine and cosine functions mentioned earlier, there are other trigonometric functions like tangent, cotangent, secant, and cosecant.

These functions are periodic, meaning they repeat their values in regular intervals, and are defined for all real numbers when considering them as functions on the unit circle. For instance, the sine and cosine functions have a period of \(2\pi\) radians, which means they repeat every \(2\pi\) radians. Knowing the behavior and graphs of these functions is beneficial, especially when dealing with problems involving angles, oscillations, and waves. Recognizing the derivatives of trigonometric functions and how they relate to each other (such as the fact that the derivative of \(\sin x\) is \(\cos x\)) is essential for solving calculus problems related to rates of change and motion.