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 $f(x)=\sin x .$ Show that $$ \frac{\Delta f}{\Delta x}=(\sin x)\left(\frac{\cos h-1}{h}\right)+(\cos x)\left(\frac{\sin h}{h}\right) $$

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 $f(x)=\sin x .$ Show that
$$
\frac{\Delta f}{\Delta x}=(\sin x)\left(\frac{\cos h-1}{h}\right)+(\cos x)\left(\frac{\sin h}{h}\right)
$$

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Key Concepts

Average Rate of Change

The average rate of change of a function over an interval is the ratio of the change in the function's value to the change in the input variable. It is calculated using the formula (f(x+h) - f(x)) / h and represents the overall change of the function with respect to x over that interval.

Difference Quotient

The difference quotient is a fundamental concept in calculus used to capture the average change of a function over a small interval. It forms the basis for the derivative, as taking the limit of the difference quotient as the interval length h approaches zero gives the instantaneous rate of change.

Trigonometric Addition Formula

The sine addition formula, sin(x+h) = sin x cos h + cos x sin h, is a key trigonometric identity used to express the sine of a sum of angles in terms of the sine and cosine of the individual angles. This formula is crucial in deriving formulas for the average rate of change of sine, cosine, and tangent functions, as it allows the separation of terms that are then factored and analyzed individually.

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