Problem 124 - Precalculus

Question

In calculus, the value \(F(b)-F(a)\) of a function \(F(x)\) at \(x=a\) and \(x=b\) plays an important role in the calculation of definite integrals. Find the exact value of \(F(b)-F(a)\) $$F(x)=\cot x-\csc ^{2} x, a=\frac{7 \pi}{6}, b=\frac{7 \pi}{4}$$

Step-by-Step Solution

Answer

The exact value of \( F(b) - F(a) \) is \( 1 - \sqrt{3} \).

Step 1: Evaluate the function at x = b

First, let's evaluate the function \( F(x) = \cot x - \csc^2 x \) at \( x = \frac{7\pi}{4} \):1. Calculate \( \cot \frac{7\pi}{4} = \frac{1}{\tan \frac{7\pi}{4}} \).2. Since \( \tan \frac{7\pi}{4} = \tan (-\frac{\pi}{4}) = -1 \), it follows that \( \cot \frac{7\pi}{4} = -1 \).3. Calculate \( \csc^2 \frac{7\pi}{4} = \frac{1}{\sin^2 \frac{7\pi}{4}} \).4. Since \( \sin \frac{7\pi}{4} = \sin (-\frac{\pi}{4}) = -\frac{\sqrt{2}}{2} \), it follows that \( \csc^2 \frac{7\pi}{4} = \left( \frac{2}{\sqrt{2}} \right)^2 = 2 \).5. Therefore, \( F(\frac{7\pi}{4}) = -1 - 2 = -3 \).

Step 2: Evaluate the function at x = a

Now, let's evaluate the function \( F(x) = \cot x - \csc^2 x \) at \( x = \frac{7\pi}{6} \):1. Calculate \( \cot \frac{7\pi}{6} = \frac{1}{\tan \frac{7\pi}{6}} \).2. Since \( \tan \frac{7\pi}{6} = \tan (\pi + \frac{\pi}{6}) = \tan \frac{\pi}{6} = \frac{1}{\sqrt{3}} \), it follows that \( \cot \frac{7\pi}{6} = \sqrt{3} \).3. Calculate \( \csc^2 \frac{7\pi}{6} = \frac{1}{\sin^2 \frac{7\pi}{6}} \).4. Since \( \sin \frac{7\pi}{6} = \sin (\pi + \frac{\pi}{6}) = -\sin \frac{\pi}{6} = -\frac{1}{2} \), it follows that \( \csc^2 \frac{7\pi}{6} = \left( -2 \right)^2 = 4 \).5. Therefore, \( F(\frac{7\pi}{6}) = \sqrt{3} - 4 \).

Step 3: Calculate the difference F(b) - F(a)

Now we can find the exact value of \( F(b) - F(a) \):\[F(\frac{7\pi}{4}) - F(\frac{7\pi}{6}) = (-3) - (\sqrt{3} - 4)\]This simplifies to:\[-3 - \sqrt{3} + 4 = 1 - \sqrt{3}\]Thus, the value of \( F(b) - F(a) \) is \( 1 - \sqrt{3} \).

Key Concepts

Cotangent Function

The **Cotangent Function**, denoted as \( \cot x \), is one of the less commonly used trigonometric functions, but it is incredibly useful in calculus and analytic trigonometry. At its core, the cotangent function is the reciprocal of the tangent function. This means:

• \( \cot x = \frac{1}{\tan x} \)

The tangent of x is usually defined as \( \frac{\sin x}{\cos x} \). Therefore, when you take the cotangent, you are essentially inverting the division of sine by cosine, resulting in:

• \( \cot x = \frac{\cos x}{\sin x} \)

Cotangent is usually undefined wherever sine is zero, as division by zero is not possible. This function can oscillate between positive and negative values, depending on the quadrant in which the angle \( x \) is located. Understanding this function is crucial when dealing with integrals, as it often pops up in the simplification of trigonometric expressions.

Cosecant Function

The **Cosecant Function**, or \( \csc x \), is yet another reciprocal trigonometric function. In this case, cosecant is the reciprocal of the sine function:

• \( \csc x = \frac{1}{\sin x} \)
• Thus, \( \csc^2 x = \left( \frac{1}{\sin x} \right)^2 \)

Cosecant is undefined wherever sine is zero, leading to its periodic undefined points. This function is important because it plays a major role in many identities and integrals. Notably, it often appears when decomposing derivatives back into their trigonometric components.
In the given problem, we see \( \csc^2 x \), which represents a squared form of the cosecant function. Knowing how to manipulate and evaluate \( \csc x \) is beneficial when calculating derivatives and integrals, as it often appears when simplifying functions.

Trigonometric Identities

**Trigonometric Identities** are a set of equations that are true for all values of the variables where both sides of the identity are defined. These identities often simplify complex trigonometric expressions and calculations. Some basic identities include:

• Pythagorean Identity: \( \sin^2 x + \cos^2 x = 1 \)
• Reciprocal Identities: For example, \( \cot x = \frac{1}{\tan x} \) and \( \csc x = \frac{1}{\sin x} \)

These identities are vital for reducing expressions involving trig functions to their simplest forms.
In calculus, particularly with definite integrals, trigonometric identities allow us to convert expressions into a more recognizable form that is easier to integrate or differentiate. For instance, recognizing that\( \csc^2 x \) as part of a derivative of a cotangent function can help in performing integration more efficiently when calculating results like \( F(b)-F(a) \). Understanding and applying these identities are key skills for solving advanced mathematical problems.