Problem 53 - Algebra and Trigonometry with Analytic Geometry

Question

If \(f(x)=\cos x\), show that $$ \frac{f(x+h)-f(x)}{h}=\cos x\left(\frac{\cos h-1}{h}\right)-\sin x\left(\frac{\sin h}{h}\right) $$

Step-by-Step Solution

Answer

The expression is shown as stated using trigonometric identities.

Step 1: Understand the function and the problem

The function given is \( f(x) = \cos x \). We need to show that the expression \( \frac{f(x+h) - f(x)}{h} \) can be rewritten as \[ \cos x\left(\frac{\cos h-1}{h}\right) - \sin x\left(\frac{\sin h}{h}\right) \] This involves evaluating the difference quotient for this function.

Step 2: Apply the trigonometric identity for \( f(x+h) \)

Using the trigonometric identity for cosine, we find: \[ f(x+h) = \cos(x+h) = \cos x \cos h - \sin x \sin h \] Substitute this into the difference quotient: \[ \frac{\cos(x+h) - \cos x}{h} = \frac{(\cos x \cos h - \sin x \sin h) - \cos x}{h} \] Simplify the expression inside the numerator.

Step 3: Simplify the numerator

Simplify the numerator from the previous step: \[ (\cos x \cos h - \sin x \sin h) - \cos x = \cos x \cos h - \cos x - \sin x \sin h \] Factor like terms: \[ \cos x (\cos h - 1) - \sin x \sin h \] This gives us the new numerator.

Step 4: Separate the terms in the difference quotient

Now substitute the simplified numerator into the difference quotient: \[ \frac{\cos x (\cos h - 1) - \sin x \sin h}{h} \] Distribute the denominator across the terms, resulting in two separate fractions: \[ \cos x \left(\frac{\cos h - 1}{h}\right) - \sin x \left(\frac{\sin h}{h}\right) \] Now, the original expression is rewritten as desired.

Key Concepts

Trigonometric Identity

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. One of the most fundamental identities used in trigonometry is the angle addition formula. For cosine, this is written as:

• \( \cos(x + h) = \cos x \cdot \cos h - \sin x \cdot \sin h \)

In our problem, we are working with the function \( f(x) = \cos x \). When tackling the difference quotient for \( f(x + h) \), applying the angle addition formula helps us express \( \cos(x + h) \) in terms of \( \cos x \) and \( \sin x \).
This is a crucial step to break down expressions in calculus problems involving trigonometric functions. This simplification allows us to manipulate and reframe the problem into a more workable form that can be solved step-by-step using algebraic techniques. Understanding and applying these identities correctly is essential for working through trigonometric problems in calculus.

Derivative of Cosine

The derivative of cosine plays a crucial role in calculus, especially when evaluating the rate of change or the slope of cosine functions. If we have a function \( f(x) = \cos x \), the derivative \( f'(x) \) is obtained using limits. However, it is essential to understand the trigonometric rules first.
The derivative of cosine is:

• \( f'(x) = -\sin x \)

This derivative is involved in many calculus problems and connecting its significance helps us transition smoothly through the steps when finding derivatives through limit processes.
In the context of our problem, the expression obtained after simplifying the difference quotient resembles the derivative form, as it involves certain components that reflect this derivative. By efficiently managing these components, you can unveil crucial insights about function behavior at specific points.

Limit Process

The limit process is a fundamental technique in calculus used to find derivatives and investigate the behavior of functions as they approach specific values. To find the derivative, often described as the slope of the tangent line, we use a limit to evaluate this slope precisely when the interval \( h \) approaches zero.

• The general form for the limit definition of a derivative is:\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]

In our specific problem, this involves evaluating the expression\[ \lim_{h \to 0} \left[ \cos x \frac{\cos h - 1}{h} - \sin x \frac{\sin h}{h} \right] \]As \( h \) approaches zero, these terms simplify using standard limits of trigonometric functions:

• \( \lim_{h \to 0} \frac{\sin h}{h} = 1 \)
• \( \lim_{h \to 0} \frac{\cos h - 1}{h} = 0 \)

These known limits help simplify expressions and provide insights into the function's behavior as we analyze the derivative step. Understanding this orderly process allows more precision and clarity in solving calculus problems.