Derivatives of sin x, cos x — proofs from first principles: Foundations

This page assumes nothing. Every symbol the parent note Derivatives of sin/cos leans on is built here, from the ground up, in an order where each idea rests only on the ones before it.

1. Angle — and why we measure it in radians

We could count this spread in degrees (a full turn = $360$ ) or in radians. The topic insists on radians, so we must understand what a radian is.

Figure — Derivatives of sin x, cos x — proofs from first principles

Look at the red arc above. Because the radius is $1$ , a full circle has circumference $2 π \cdot 1 = 2 π$ , so a full turn is $2 π$ radians. The symbol $π$ (pi) is just the number $\approx 3.14159$ — the ratio of any circle's circumference to its diameter.

2. $sin$ and $cos$ — heights and widths on the circle

The red vertical segment is $sin x$ ; the black horizontal segment is $cos x$ . As you sweep $x$ around, $P$ traces the circle, and its height $sin x$ goes up and down like a wave.

Signs in every quadrant — we must cover all cases, because $P$ visits all four:

Quadrant	angle range	$cos x$ (horizontal)	$sin x$ (vertical)
I	$0$ to $\frac{π}{2}$	$+$	$+$
II	$\frac{π}{2}$ to $π$	$-$	$+$
III	$π$ to $\frac{3 π}{2}$	$-$	$-$
IV	$\frac{3 π}{2}$ to $2 π$	$+$	$-$

3. The function symbol $f$ and $f (x)$

When we write $f (x + h)$ we mean: feed the machine the number $x + h$ instead of $x$ . So $sin (x + h)$ is the height at a slightly later angle. Nothing more mysterious than that.

4. Limits — the "sneak up on it" symbol

Why do we need this? Because slope needs two points, but a derivative wants the slope at one instant. We take two points a distance $h$ apart and let $h$ shrink toward $0$ — but we can't set $h = 0$ (that gives $\frac{0}{0}$ , meaningless). The limit is the honest way to ask "what happens as the gap vanishes?" See Limits — definition and properties.

5. The derivative — slope of a curve at a point

The two black dots are $(x, f (x))$ and $(x + h, f (x + h))$ . The rise is $f (x + h) - f (x)$ , the run is $h$ , so the slope of the line joining them (the red line) is $\frac{f ( x + h ) - f ( x )}{h} .$ As $h$ shrinks, that red line pivots toward the true tangent. Taking the limit gives the first-principles derivative: $f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h} .$ The symbols: $f^{'} (x)$ ("f prime of x") and $\frac{d}{d x}$ both mean "the derivative of." $\frac{d}{d x} sin x$ = "the rate at which $sin x$ changes as $x$ changes."

6. The addition formulas — the algebraic splitter

Why does the topic need these? The derivative feeds the machine $x + h$ . To take the limit we must separate the part that depends on $x$ (which stays fixed) from the part that depends on $h$ (which is shrinking). These formulas do exactly that surgery — they peel $sin (x + h)$ into a $sin x$ chunk and a $cos x$ chunk, each multiplied by something in $h$ alone. Full derivation lives in Trigonometric addition formulas.

7. The Squeeze Theorem — trapping a limit

Picture a coin squeezed between two hands closing together; the coin has nowhere to go but where the hands meet. The parent traps $\frac{h}{s i n h}$ between $1$ and $\frac{1}{c o s h}$ using three circle areas. Details in Squeeze Theorem.

The prerequisite map

Equipment checklist

Recall Test your readiness (reveal each answer)

What is one radian, in one sentence? ::: The angle whose arc on the unit circle has length equal to the radius ( $= 1$ ). On the unit circle, $sin x$ is the ___ of the point and $cos x$ is the ___. ::: height (vertical position); width (horizontal position) In which quadrants is $sin x$ negative? ::: III and IV (the lower half). Write the first-principles derivative of $f$ . ::: $f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}$ Why can't we just set $h = 0$ in the slope quotient? ::: It gives $\frac{0}{0}$ , which is undefined — the limit sneaks up instead. State the sine addition formula. ::: $sin (x + h) = sin x cos h + cos x sin h$ State the cosine addition formula. ::: $cos (x + h) = cos x cos h - sin x sin h$ What does the Squeeze Theorem let you conclude? ::: If $L \leq M \leq U$ and both $L, U \to$ the same value, then $M$ heads there too. Why radians and not degrees for calculus? ::: Only in radians does a tiny angle equal its arc, making $\frac{s i n h}{h} \to 1$ clean. What identity turns $cos^{2} h - 1$ into $- sin^{2} h$ ? ::: $cos^{2} h + sin^{2} h = 1$ (Pythagoras on the unit circle).