Derivatives of all six trig functions: Foundations

This page assumes you have seen nothing. Before you touch the parent note the six trig derivatives, you must be fluent with every squiggle it writes. We build them one at a time, each on top of the last.

1. The unit circle — the stage everything sits on

Everything in trigonometry lives on one circle of radius 1, centred at the origin. We call it the unit circle because "unit" just means "one".

Figure — Derivatives of all six trig functions

Why radians and not degrees? Because on the unit circle the angle equals the arc length only in radians. That single fact is what makes $\frac{sin h}{h} \to 1$ work later. Degrees would smuggle in an ugly factor of $\frac{π}{180}$ . This is the "silent killer" the parent warns about — and it starts here, in the definition.

2. $sin$ , $cos$ , and the point on the circle

Now stand the point on the circle and drop it onto the two axes.

Why these two numbers matter so much. Every derivative in the parent takes a limit as the step $h \to 0$ , which drives the little angle $h$ toward $0$ . At that moment $cos h \to cos 0 = 1$ and $sin h \to sin 0 = 0$ . Those are precisely the values that collapse the algebra — remember them; they are the landing spot of every trig limit.

Because the point sits on a circle of radius 1, the horizontal and vertical legs and the radius form a right triangle with legs $cos θ$ , $sin θ$ and hypotenuse $1$ . Pythagoras then gives us, for free, the identity: $sin^{2} θ + cos^{2} θ = 1.$ Here $sin^{2} θ$ is shorthand for $(sin θ)^{2}$ — the number $sin θ$ squared, not "sin of $θ^{2}$ ". Keep that distinction; the parent uses it constantly.

2b. Negative angles — the mirror symmetry

Because a limit $h \to 0$ lets $h$ arrive from below zero too (a small negative nudge, a clockwise rotation), we must know what $sin$ and $cos$ do for negative angles.

Why we need this. In $lim_{h \to 0}$ the step $h$ approaches from the positive side and the negative side. The ratio $\frac{sin h}{h}$ must head to the same value from both. Because $sin (- h) = - sin h$ and the bottom also flips to $- h$ , the two minus signs cancel: $\frac{sin ( - h )}{- h} = \frac{sin h}{h}$ . The limit is genuinely two-sided, and this symmetry is why both sides agree.

3. The four ratio functions — $tan, cot, sec, csc$

The other four trig functions are not new geometry. They are $sin$ and $cos$ divided by each other. This is why the whole topic reduces to two facts.

4. The limit symbol $h \to 0 lim$

The parent's first move is the limit definition of the derivative. You must read the symbol before that sentence makes sense.

Why we need a limit at all. Slope means "rise over run", $\frac{change in height}{change in x}$ . But the instantaneous slope needs the run to be zero — and $\frac{0}{0}$ is meaningless. The limit is the honest way to ask "what would the ratio be if the run could shrink to nothing?" without ever dividing by a true zero. That is the entire engine of Limit definition of the derivative.

5. The derivative symbol $\frac{d}{d x}$

Every derivation in the parent begins by writing exactly this line, then replacing $f$ with $sin$ or $cos$ .

6. Angle addition — how to expand $sin (x + h)$

The step $h$ gets added inside the angle, giving $sin (x + h)$ . To make progress we must split that apart.

Why this tool and not another? The derivative forces the combination $x + h$ upon us. The only way to separate the fixed part $x$ from the shrinking part $h$ — so the limit can act on $h$ alone — is this identity. It is the hinge of Steps 1 and 2 in the parent. Notice: once split, the $h$ -pieces are exactly $cos h$ and $sin h$ , which head to $cos 0 = 1$ and $sin 0 = 0$ — the values from §2.

7. The rules that recombine everything

Once $sin^{'}$ and $cos^{'}$ are known, three algebra rules finish the job. You only need to recognise their shapes here. In each rule below, $u$ and $v$ are just names for two smaller functions you have glued together — on this page they will stand for $sin x$ and $cos x$ (or the constant $1$ ). Think of $u$ and $v$ as "the top piece" and "the bottom piece".

The quotient rule is the workhorse, because each of $tan, cot, sec, csc$ is a fraction of $sin$ and $cos$ — so $u$ and $v$ become $sin x$ and $cos x$ .
The chain rule appears the moment something sits inside the trig function, like $sec (3 x)$ — the inner $3 x$ must be differentiated too.

8. How it all fits together

The circle gives us $sin, cos$ and their starting values; those feed the two limits; the limits give $sin^{'}$ and $cos^{'}$ ; the three algebra rules then spread those two facts to all six functions.

Equipment checklist

What is

π

, in one phrase?

The arc length of a half-lap around the unit circle, about

3.14159

What does a radian measure on the unit circle?

The arc length the angle cuts — so angle equals arc length when radius is 1

What are the coordinates of the unit-circle point at angle

θ

(cos θ, sin θ)

— width then height

What are

sin 0

and

cos 0

sin 0 = 0

and

cos 0 = 1

(point sits at the far right)

What are the symmetry rules for a negative angle?

cos (- θ) = cos θ

and

sin (- θ) = - sin θ

Why do we care about negative angles here?

A two-sided limit lets

h \to 0

from below, so we need

sin, cos

of small negative angles

sin^{2} θ

the same as

sin (θ^{2})

No —

sin^{2} θ = (sin θ)^{2}

, the value squared

Write

tan, cot, sec, csc

as ratios of

sin

and

cos

tan = \frac{s i n}{c o s}

cot = \frac{c o s}{s i n}

sec = \frac{1}{c o s}

csc = \frac{1}{s i n}

Where is

sec θ

undefined?

Wherever

cos θ = 0

, i.e.

θ = \frac{π}{2}, \frac{3 π}{2}, \dots

In plain words, what does

lim_{h \to 0} F (h)

ask?

What single value

F (h)

approaches as

h

shrinks toward

0

from both sides

What are the two foundation limits?

lim_{h \to 0} \frac{s i n h}{h} = 1

and

lim_{h \to 0} \frac{c o s h - 1}{h} = 0

What do

\frac{d}{d x} f

and

f^{'} (x)

both mean?

The slope of

f

at each point

x

(they are the same thing)

State the limit definition of the derivative.

f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}

Expand

sin (x + h)

sin x cos h + cos x sin h

In the quotient rule, what do

u

and

v

stand for here?

The top and bottom pieces — e.g.

u = sin x

v = cos x

Which rule turns

sin, cos

derivatives into the other four?

The quotient rule, since each is a ratio of

sin

and

cos

State the Pythagorean identity.

sin^{2} θ + cos^{2} θ = 1

Connections

Derivatives of all six trig functions (the parent this page prepares you for)
Limit definition of the derivative
The Squeeze Theorem
Angle addition formulas
Pythagorean identity sin^2 + cos^2 = 1
Quotient Rule
Chain Rule
Product Rule
Simple Harmonic Motion
Derivatives of inverse trig functions