Unit circle definition of trig functions — all 6 trig functions for any angle: Foundations

This page assumes you have seen none of the notation in the parent topic. We build each symbol from a picture, in an order where every new symbol only uses ones already earned. If a line here feels obvious, good — the parent note leans on all of it.

0. What a "coordinate" even is

Before circles, before angles, we need to be able to point at a spot on a flat page and say exactly where it is with two numbers.

Figure — Unit circle definition of trig functions — all 6 trig functions for any angle

Look at the figure. The plum point sits $2$ steps right and $1$ step up, so its address is $(2, 1)$ . Negative numbers just mean the other direction: left for a negative first number, down for a negative second.

Reveal-test:

The point

(- 3, 4)

is where?

3

steps left,

4

steps up.

1. Distance, and the number $x^{2} + y^{2}$

Now: how far is a point from the origin?

Why squared and not just distance? Because of the right triangle in the figure. The point $(x, y)$ , the origin, and the foot on the x-axis form a right-angled triangle with legs of length $∣ x ∣$ and $∣ y ∣$ . The Pythagorean theorem — the rule that in any right triangle the two short sides squared add up to the long side squared — says the straight-line distance $r$ satisfies

$r^{2} = x^{2} + y^{2} .$

We keep it squared because that avoids square roots and because "on a circle of radius 1" becomes the very clean statement $x^{2} + y^{2} = 1$ . This exact equation is where the parent's Pythagorean identity comes from — see Pythagorean identities.

Why radius 1 and not any other number? Because dividing distances by the radius (to compare ratios) is trivial when the radius is $1$ — you divide by nothing. This is the whole reason the parent can say " $cos θ = x$ " with no fraction.

2. Angle $θ$ — a rotation, not a triangle

The symbol $θ$ (Greek letter "theta") is the parent topic's main character. It looks scary; it is not.

Read the figure like a clock that spins backwards:

$θ = 0°$ : you are at $(1, 0)$ , the far right.
$θ = 90°$ : a quarter turn, now at the top $(0, 1)$ .
$θ = 180°$ : half turn, at the far left $(- 1, 0)$ .
$θ = 270°$ : three-quarter turn, at the bottom $(0, - 1)$ .

Negative and huge angles, previewed:

Negative $θ$ = spin the other way (clockwise). $- 90°$ lands you at the bottom $(0, - 1)$ .
$θ$ bigger than $360°$ = you looped past the start. $370°$ is the same landing spot as $10°$ .

This is why the parent calls the functions "periodic" — see Periodicity and $2\pi$. Two units for $θ$ exist; the arc-length one lives in Radian measure.

The four quadrants (Q1 top-right, Q2 top-left, Q3 bottom-left, Q4 bottom-right) are just the four regions you pass through as $θ$ grows. The parent's sign table is a direct read-off of which of $x, y$ are positive in each.

Reveal-test:

Where does

θ = 180°

land you?

(- 1, 0)

, the far left of the circle.

3. The two legs on the circle — meeting $cos$ and $sin$ at their root

The parent already knows Right-triangle trigonometry (SOH-CAH-TOA). Here is the bridge, built symbol by symbol.

SOH-CAH-TOA is a memory chant for three ratios:

Sine = Opposite / Hypotenuse $= \frac{y}{1} = y$ .
Cosine = Adjacent / Hypotenuse $= \frac{x}{1} = x$ .
Tangent = Opposite / Adjacent $= \frac{y}{x}$ .

Because the hypotenuse is $1$ , the sine and cosine shed their fractions and become just $y$ and $x$ . That is the entire reason the parent can write $cos θ = x$ and $sin θ = y$ with a straight face.

4. Ratios, reciprocals, and division-by-zero

The parent's other four functions are just arithmetic on $x$ and $y$ . To read them you need two ideas.

Now the parent's four extras are transparent:

Why do some blow up? Whenever a coordinate hits zero, any function that divides by it becomes undefined:

$x = 0$ (at $θ = 90°, 270°$ ): $tan$ and $sec$ die (they divide by $x$ ).
$y = 0$ (at $θ = 0°, 180°$ ): $cot$ and $csc$ die (they divide by $y$ ).

This is not a special rule to memorise — it is the same division-by-zero ban you just learned, applied on the circle.

5. Even and odd — a mirror trick for negative angles

One last pair of words the parent leans on.

Picture the negative angle as walking clockwise instead. You land at the mirror image across the x-axis: same $x$ , opposite $y$ . So:

Cosine is even — the $x$ is unchanged: $cos (- θ) = cos θ$ .
Sine is odd — the $y$ flips: $sin (- θ) = - sin θ$ .

More on this classification lives in Even and odd functions; the parent uses it to handle negative angles instantly.

How it all feeds the topic

See it flow into Unit circle definition of trig functions — all 6 trig functions for any angle.

Equipment checklist

Test yourself — cover the right side.

Read off the address of a point

3

left and

2

down.

(- 3, - 2)

Why is

x^{2}

never negative?

A number times itself; negative

\times

negative

=

positive.

State the Pythagorean theorem for legs

a, b

and hypotenuse

c

a^{2} + b^{2} = c^{2}

What equation describes the unit circle?

x^{2} + y^{2} = 1

What does

θ

measure, and from where?

Rotation counterclockwise, starting at

(1, 0)

Where do you land at

θ = 270°

(0, - 1)

, the bottom.

On the circle triangle, which leg is "adjacent"?

The horizontal one along the x-axis, length

x

Turn SOH-CAH-TOA into unit-circle form.

sin θ = y

cos θ = x

tan θ = y / x

What is the reciprocal of

x

1/ x

Why is

\frac{4}{0}

undefined?

Division by zero has no answer.

Even or odd: cosine?

Even —

cos (- θ) = cos θ

Even or odd: sine?

Odd —

sin (- θ) = - sin θ

Connections

Right-triangle trigonometry (SOH-CAH-TOA) — the source of the three base ratios.
Pythagorean identities — grows straight out of $x^{2} + y^{2} = 1$ .
Radian measure — the other way to size $θ$ .
Periodicity and $2\pi$ — why looping repeats the answer.
Even and odd functions — the mirror trick for negative angles.
Reference angles — reducing big angles to acute ones (next tools).
Graphs of trig functions — plotting these coordinates against $θ$ .