You are asked to derive expressions for the average rates of...

You are asked to derive expressions for the average rates of change of the functions $\sin x, \cos x,$ and $\tan x$ In each case, assume that the interval is $[x, x+h] .$ (The results are used in calculus in the study of derivatives.) Let $g(x)=\cos x .$ Show that $$ \frac{\Delta g}{\Delta x}=(\cos x)\left(\frac{\cos h-1}{h}\right)-(\sin x)\left(\frac{\sin h}{h}\right) $$

You are asked to derive expressions for the average rates of change of the functions $\sin x, \cos x,$ and $\tan x$
In each case, assume that the interval is $[x, x+h] .$ (The results are used in calculus in the study of derivatives.)
Let $g(x)=\cos x .$ Show that
$$
\frac{\Delta g}{\Delta x}=(\cos x)\left(\frac{\cos h-1}{h}\right)-(\sin x)\left(\frac{\sin h}{h}\right)
$$

Key Concepts

Derivatives of Trigonometric Functions

Derivatives of trigonometric functions, such as the derivative of cosine being -sin x, are derived using the difference quotient and limit definitions. Understanding how to manipulate and simplify expressions involving trigonometric functions and their limits, such as the limits involving (cos h - 1)/h and sin h/h, is crucial in establishing these standard derivative results.

Limit Process in Calculus

The limit process involves examining the behavior of a function as the change in the input variable, h, approaches zero. In the context of the difference quotient, limits are used to transition from an average rate of change over an interval to the instantaneous rate of change, or derivative, thereby linking the two fundamental concepts in calculus.

Difference Quotient

The difference quotient, defined as (f(x+h) - f(x)) / h for a function f, represents the average rate of change over an interval [x, x+h]. It is the foundational concept in calculus that leads to the definition of the derivative through the limit process as h approaches zero.

Trigonometric Addition Formulas

Trigonometric addition formulas, such as the one for cosine (cos(a+b) = cos a cos b - sin a sin b), are essential tools for expanding expressions involving trigonometric functions. They allow us to express the value of a trigonometric function at a sum of angles in terms of the functions at individual angles, which is key for deriving average rate of change expressions for functions like cosine.

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