Problem 48 - Algebra and Trigonometry

Question

Let \(g(x)=\cos x .\) Show that \(\frac{g(x+h)-g(x)}{h}=-\cos x\left(\frac{1-\cos h}{h}\right)-\sin x\left(\frac{\sin h}{h}\right)\)

Step-by-Step Solution

Answer

The given result is achieved by using trigonometric identities and rearranging the terms.

Step 1: Define the Difference Quotient

Start by writing the difference quotient for the function \(g(x) = \cos x\), which is \( \frac{g(x+h) - g(x)}{h} = \frac{\cos(x+h) - \cos x}{h} \).

Step 2: Use the Angle Addition Formula

Recall the trigonometric identity for the cosine of a sum: \( \cos(x+h) = \cos x \cos h - \sin x \sin h \). Substitute this into the difference quotient to get \( \frac{\cos x \cos h - \sin x \sin h - \cos x}{h} \).

Step 3: Reorganize the Expression

Regroup the terms as \( \frac{\cos x \cos h - \cos x - \sin x \sin h}{h} = \frac{\cos x (\cos h - 1) - \sin x \sin h}{h} \).

Step 4: Split the Fraction

Separate the fraction into two parts: \( \frac{\cos x (\cos h - 1)}{h} - \frac{\sin x \sin h}{h} \).

Step 5: Factor Out Constants

Factor out \( \cos x \) from the first term and \( \sin x \) from the second term to get: \( \cos x \left( \frac{(\cos h - 1)}{h} \right) - \sin x \left( \frac{\sin h}{h} \right) \).

Step 6: Adjust Terms to Match Given Expression

Notice that the expression is almost the given solution, but we need to multiply by \(-1\) to match completely. Therefore, rearrange and multiply by \(-1\) to obtain the expression: \(-\cos x \left( \frac{1 - \cos h}{h} \right) - \sin x \left( \frac{\sin h}{h} \right)\).

Key Concepts

Limit Process

The limit process is an essential mathematical concept used in calculus to find the value that a function approaches as the input approaches a particular point. In the context of difference quotients, limits help articulate the derivative of a function. When dealing with the function \(g(x) = \cos x\), we use the difference quotient \( \frac{g(x+h) - g(x)}{h} \). As \(h\) approaches zero, this quotient tends to the derivative of \(g(x)\) at the point \(x\). The derivative represents the rate of change or the slope of the tangent line to the curve at any given point. Calculating the limit as \(h\) approaches zero involves:

• Simplifying the algebraic expression via identities or formulas, such as trigonometric or limit identities.
• Using these simplifications to reduce terms that might become undefined or indeterminate forms like \(\frac{0}{0}\).
• Using L'Hôpital's Rule if necessary for indeterminate forms or simply applying known limits such as \(\lim_{h \to 0} \frac{\sin h}{h} = 1\) and \(\lim_{h \to 0} \frac{1-\cos h}{h} = 0\).

Understanding the limit process is crucial not just for solving calculus problems but also for grasping concepts like continuity, differentiability, and integrals. Remember, in the given problem, applying the limit helps us confirm the derivative constructed from the difference quotient.

Trigonometric Identities

Trigonometric identities are crucial facilitators when it comes to simplifying expressions or transforming them into equivalent forms. These identities express relationships between the trigonometric functions and angles. In the exercise, the cosine addition formula plays a major role. The identity for the cosine of a sum is:\[ \cos(x+h) = \cos x \cos h - \sin x \sin h. \]This identity lets us break down complex expressions into simpler components by expressing new angles as sums of simpler angles. Here are some common trigonometric identities you might find useful:

• Basic identities: \( \tan x = \frac{\sin x}{\cos x} \), \(1 + \tan^2 x = \sec^2 x\).
• Even-Odd identities: \( \sin(-x) = -\sin x \), \( \cos(-x) = \cos x \).
• Complementary: \( \sin(90^\circ - x) = \cos x \), \( \cos(90^\circ - x) = \sin x \).

In essence, deep diving into these identities can make solving trigonometric equations or simplifying expressions more manageable and intuitive. This is why they are so important for calculus, as seen in this exercise.

Angle Addition Formula

The angle addition formula is a fundamental trigonometric identity used to simplify expressions or solve problems involving angles. In the exercise, we specifically focus on the cosine angle addition formula:\[ \cos(x+h) = \cos x \cos h - \sin x \sin h. \]This formula allows us to express the cosine of a sum of two angles \(x + h\) in terms of the cosine and sine of the individual angles. It can be particularly useful when evaluating limits or dealing with the derivatives of trigonometric functions.Using the angle addition formula aids in transforming the original problem into a more workable form, as it allows us to rearrange the terms of the trigonometric expression:

• It simplifies the calculation by expressing new angles in terms of known trigonometric values.
• Helps us avoid direct limit computations of complex trigonometric expressions.
• Makes it easier to identify terms that can be grouped or factored for further simplification.

Thus, the angle addition formula is not just a handy tool in this exercise, but a central concept for manipulating trigonometric expressions, essential to succeeding in calculus. By mastering such fundamental identities, tackling derivative problems involving trigonometric functions becomes significantly more straightforward.