Problem 46 - A Textbook of Differential Calculus

Question

Statement I The range of \(f(x)=\sin \left(\frac{\pi}{5}+x\right)-\sin \left(\frac{\pi}{5}-x\right)-\sin \left(\frac{2 \pi}{5}+x\right)\) \(+\sin \left(\frac{2 \pi}{5}-x\right)\) is \([-1,1]\) Statement II \(\cos \frac{\pi}{5}-\cos \frac{2 \pi}{5}=\frac{1}{2}\).

Step-by-Step Solution

Answer

Both statements are true; Statement II explains the process needed for Statement I.

Step 1: Identify the Required Range

To find the range of the function, we need to understand the behavior of the terms involving sine. Recall that the sine function, \(\sin(x)\), has the range of \([-1, 1]\).

Step 2: Use Sine Addition Formulas

Each term of \(f(x)\) involves expressions of the form \(\sin(a \pm x)\). Using the sine addition formula \(\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B\), we can transform the function to see any simplifications or patterns.

Step 3: Simplify the Function

Apply the sine formula to transform \(f(x)\):- \( \sin\left(\frac{\pi}{5} + x\right) = \sin\frac{\pi}{5}\cos x + \cos\frac{\pi}{5}\sin x \)- \( \sin\left(\frac{\pi}{5} - x\right) = \sin\frac{\pi}{5}\cos x - \cos\frac{\pi}{5}\sin x \)- \( \sin\left(\frac{2\pi}{5} + x\right) = \sin\frac{2\pi}{5}\cos x + \cos\frac{2\pi}{5}\sin x \)- \( \sin\left(\frac{2\pi}{5} - x\right) = \sin\frac{2\pi}{5}\cos x - \cos\frac{2\pi}{5}\sin x \)Combine similar terms for \(f(x)\) to see the transformations.

Step 4: Compute Using Trigonometric Identities

After applying the formulas, observe if terms simplify by trigonometric identities. For this exercise, upon simplification, you'll find that terms cancel:\[f(x) = 2\sin x \left(\cos\frac{\pi}{5} - \cos\frac{2\pi}{5} \right) = 2\sin x \cdot \frac{1}{2} = \sin x\]Where the value \(\cos \frac{\pi}{5} - \cos \frac{2\pi}{5} = \frac{1}{2}\) is used (as given in Statement II).

Step 5: Conclusion on the Range of f(x)

The simplified equation \(f(x) = \sin x\) shows that the range of \(f(x)\) is the range of \(\sin x\), which is \([-1, 1]\). Hence, Statement I is true.Statement II is also verified true since it was used in the simplification.

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical tools that help simplify and solve problems involving trigonometric functions like sine, cosine, and tangent. Understanding these identities is crucial for working with functions and solving equations efficiently. One commonly used identity in differential calculus and trigonometry is the Sine Addition Formula, which states:

• \( \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \).

These identities allow the transformation of expressions into simpler forms, making it easier to identify patterns or cancel terms. By applying these formulas, complex expressions become manageable, and their behavior can be better understood. For instance, in our example exercise, the application of the Sine Addition Formula simplifies a complex function into a basic sine expression, \( \sin x \), whose properties are well known.

Sine Function

The sine function, denoted as \( \sin(x) \), is a fundamental trigonometric function. It describes a smooth, periodic oscillation and is one of the primary building blocks of trigonometry.

• The sine function originates from the sine of an angle in a right triangle, where it represents the ratio of the length of the opposite side to the hypotenuse, for a given angle.
• One critical feature of \( \sin(x) \) is its range and periodicity. It repeats every \( 2\pi \) interval and oscillates between -1 and 1.

The function exhibits symmetric properties and has applications spanning various fields from mathematics to physics. In the context of this exercise, understanding the range and properties of the sine function directly impacts the ability to determine the range of the entire function \( f(x) \). Once the transformed function simplifies to \( \sin(x) \), predicting the range becomes as straightforward as understanding the range of \( \sin(x) \) itself.

Range of Functions

When analyzing functions, determining the range is crucial. The range of a function refers to all possible output values the function can produce. For trigonometric functions like the sine function, the range is often bounded due to their periodic nature. Let’s explore the steps to find the range of a trigonometric function:

• Firstly, identify any transformation or simplification that can be applied to the function to highlight key trigonometric terms.
• Apply trigonometric identities, like the Sine Addition Formula, to transform complex terms inside the function.
• Simplify the expression to its most basic form, in our exercise the function simplified to \( \sin(x) \). This simplification helps directly find the range.
• Based on the properties of \( \sin(x) \), determine its potential output values, which fall between -1 and 1.

Understanding these steps and the concept of range allows us to see beyond complex expressions and evaluate the behavior and limits of functions effectively. In the example provided, by knowing the range of \( \sin(x) \), we establish the range of \( f(x) \) as \([-1, 1]\), verifying Statement I.