Trigonometric Functions and the Unit Circle

The trigonometric functions (sine, cosine, tangent, and their reciprocals) are defined by the coordinates of a point traveling around the unit circle — a circle of radius 1 centered at the origin. For an angle $θ$ measured counterclockwise from the positive $x$ -axis, $cos θ$ is the $x$ -coordinate and $sin θ$ is the $y$ -coordinate of the point on the circle. This framework extends right-triangle trigonometry to all real angles, produces periodic functions, and underlies the analysis of waves, oscillations, and rotation. Developed across millennia from Hipparchus and the Indian astronomers to Euler, who unified it analytically in the 1700s.

A trigonometric function assigns to an angle a ratio derived from a right triangle, or equivalently a coordinate of a point on the unit circle. Precisely, if a point $P$ moves around the unit circle so that the arc/angle swept from $(1, 0)$ is $θ$ , then $P = (cos θ, sin θ)$ . These functions matter because they convert rotation and angle into number, making them the natural language of anything that repeats — sound, light, alternating current, planetary motion, and the components of vectors. The surprising fact is that the same six functions born from surveying triangles turn out, via Euler's formula $e^{i θ} = cos θ + i sin θ$ , to be the real and imaginary parts of the exponential function — sewing trigonometry, geometry, and calculus into a single fabric.

The Core Insight

A circle of radius 1 turns angle into position. Because the radius is exactly 1, the Pythagorean relation $x^{2} + y^{2} = 1$ becomes $cos^{2} θ + sin^{2} θ = 1$ for free, and every trigonometric identity is just geometry of that one circle read off in coordinates. Right-triangle definitions are the special case where $θ$ lives in the first quadrant; the unit circle generalizes them to negative angles, obtuse angles, and angles beyond a full turn.

Formal Statement

Radian measure

The radian measure of an angle is the length of the arc it subtends on the unit circle. A full revolution is the full circumference $2 π$ , so

36 0^{\circ} = 2 π rad, θ_{rad} = θ_{deg} \cdot \frac{π}{180} .

⟳arc length

s = r θ

with

r = 1

gives

s = θ

; the whole circle has

s = 2 π

Unit-circle trigonometric functions

Let $θ \in R$ be an angle measured counterclockwise (positive) from the positive $x$ -axis. Let $P = (x, y)$ be the point reached on the unit circle $x^{2} + y^{2} = 1$ . Define

cos θ = x, sin θ = y, tan θ = \frac{y}{x} = \frac{sin θ}{cos θ} (x \neq = 0) .

⟳project

P

onto each axis;

tan θ

is the slope of the radius

O P

The reciprocal functions are

csc θ = \frac{1}{sin θ}, sec θ = \frac{1}{cos θ}, cot θ = \frac{cos θ}{sin θ},

each defined wherever its denominator is nonzero.

⟳invert sine, cosine, tangent respectively; restrict domains to avoid division by zero.

Here $θ$ is the angle in radians, $x = cos θ$ and $y = sin θ$ are the coordinates of $P$ , and $O = (0, 0)$ is the center.

The unit circle: the point P=(cosθ, sinθ) at angle θ

Pythagorean identity

For every real $θ$ ,

cos^{2} θ + sin^{2} θ = 1.

⟳

P = (cos θ, sin θ)

lies on

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

by definition.

Dividing by $cos^{2} θ$ and by $sin^{2} θ$ gives the companion forms

1 + tan^{2} θ = sec^{2} θ, 1 + cot^{2} θ = csc^{2} θ .

⟳divide [4] through by

cos^{2} θ

(resp.

sin^{2} θ

) and apply [2]–[3].

Periodicity, parity, and co-function relations

The functions repeat and reflect according to the geometry of the circle:

sin (θ + 2 π) = sin θ, cos (θ + 2 π) = cos θ, tan (θ + π) = tan θ .

⟳adding

2 π

returns

P

to the same point;

tan

repeats every half-turn because

(x, y)

and

(- x, - y)

share a slope.

cos (- θ) = cos θ (even), sin (- θ) = - sin θ (odd) .

⟳reversing rotation reflects

P

across the

x

-axis, flipping the sign of

y

only.

sin (\frac{π}{2} - θ) = cos θ, cos (\frac{π}{2} - θ) = sin θ .

⟳reflecting

θ

\frac{π}{2} - θ

swaps the

x

- and

y

-coordinates of

P

Why It Holds / Derivation

From right triangle to circle, and the addition formula

Step 1 — Right triangle case. In a right triangle with hypotenuse 1 and acute angle

θ

, the side opposite

θ

has length

sin θ

and the adjacent side

cos θ

(definitions of opposite/hypotenuse, adjacent/hypotenuse). Placing the triangle with its right angle on the

x

-axis and vertex at

O

puts the far vertex at exactly

(cos θ, sin θ)

. Step 2 — Extension. For any real

θ

, define

P

by rotating the point

(1, 0)

through angle

θ

. The coordinates of

P

agree with Step 1 in the first quadrant and extend the functions everywhere else, automatically supplying correct signs in each quadrant. Step 3 — Pythagorean identity. Since

P

lies on the circle,

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

, i.e. equation [4]. Step 4 — Addition formula. Rotating by

α

then

β

equals rotating by

α + β

. A

2 \times 2

rotation matrix sends

(1, 0)

(cos θ, sin θ)

, and composing two rotations multiplies the matrices:

(cos β sin β - sin β cos β) (cos α sin α) = (cos (α + β) sin (α + β)) .

⟳matrix multiply the left side and read off components.

Reading off the two coordinates yields the addition formulas

sin (α + β) = sin α cos β + cos α sin β, cos (α + β) = cos α cos β - sin α sin β .

⟳equate components in [9].

■

Leonhard Euler | 1707–1783

Euler reshaped trigonometry from a table-driven craft into analysis. He standardized the notation $sin$ , $cos$ , $tan$ as functions of a numerical (radian) argument rather than line segments in a circle, and proved the identity that bears his name, $e^{i θ} = cos θ + i sin θ$ . From this single equation every addition formula, derivative, and series expansion of the trigonometric functions follows mechanically. His Introductio in analysin infinitorum (1748) is essentially the birth certificate of the modern viewpoint adopted here.

Building Intuition

Picture a clock hand of length 1 sweeping counterclockwise. Its shadow on the floor (the $x$ -axis) is $cos θ$ ; its height above the floor (the $y$ -axis) is $sin θ$ . As the hand spins, the height rises and falls smoothly between $+ 1$ and $- 1$ — that wobble is the sine wave. Cosine is the same wobble started a quarter-turn earlier (equation [8]). Tangent is the slope of the hand; it blows up to $\pm \infty$ when the hand points straight up or down because a vertical line has undefined slope. Everything "trigonometric" is just describing where the tip of that spinning hand is.

Worked Examples

Evaluating at a special angle

Given: Find

sin θ

cos θ

tan θ

for

θ = \frac{5 π}{6}

(i.e.

15 0^{\circ}

). Step 1.

15 0^{\circ}

is in the second quadrant, where

x < 0

and

y > 0

, so cosine is negative and sine is positive. Step 2. Reference angle:

18 0^{\circ} - 15 0^{\circ} = 3 0^{\circ} = \frac{π}{6}

. The known values are

sin 3 0^{\circ} = \frac{1}{2}

cos 3 0^{\circ} = \frac{3}{2}

. Step 3. Apply quadrant signs:

sin \frac{5 π}{6} = + \frac{1}{2}, cos \frac{5 π}{6} = - \frac{3}{2} .

Step 4. Tangent:

tan θ = \frac{sin θ}{cos θ} = \frac{1/2}{- 3 /2} = - \frac{1}{3}

sin \frac{5 π}{6} = \frac{1}{2}, cos \frac{5 π}{6} = - \frac{3}{2}, tan \frac{5 π}{6} = - \frac{1}{3} \approx - 0.577

(dimensionless ratios)

Modeling a Ferris wheel

Given: A Ferris wheel of radius

R = 15 m

has its center

18 m

above the ground. It rotates once every

40 s

. A rider boards at the lowest point (

t = 0

). Find the rider's height

h (t)

and their height at

t = 10 s

. Step 1. Angular speed:

ω = \frac{2 π}{40} = \frac{π}{20} rad/s

. Step 2. Starting at the bottom, height is lowest at

t = 0

, so use a downward-shifted cosine:

h (t) = 18 - 15 cos (\frac{π}{20} t) .

⟳center height minus vertical projection

R cos (ω t)

; at

t = 0

this gives

18 - 15 = 3

m, the lowest point.

Step 3. Evaluate at

t = 10 s

\frac{π}{20} \cdot 10 = \frac{π}{2}

, and

cos \frac{π}{2} = 0

h (10) = 18 - 15 \cdot 0 = 18 m .

h (t) = 18 - 15 cos (\frac{π}{20} t) m, h (10) = 18 m

Common Misconceptions

Degrees vs. radians. Calculus formulas such as

\frac{d}{d θ} sin θ = cos θ

and the limit

lim_{θ \to 0} \frac{s i n θ}{θ} = 1

hold only in radians. If your calculator is in degree mode, derivatives pick up a hidden factor of

\frac{π}{180}

. Always check the angle unit before computing. $sin^{2} θ$ is not $sin (θ^{2})$ and $sin^{- 1}$ is not $1/ sin$ . The notation

sin^{2} θ

means

(sin θ)^{2}

, but

sin^{- 1} θ

means the inverse function

arcsin θ

, not

csc θ = 1/ sin θ

. These clashing conventions are a frequent source of error.

One might guess $sin (α + β) = sin α + sin β$ . Test $α = β = \frac{π}{2}$ :

sin (π) = 0, but sin \frac{π}{2} + sin \frac{π}{2} = 1 + 1 = 2.

Since

0 \neq = 2

, sine is not additive; the correct rule is the addition formula [10].

Connections

The unit circle is where several threads meet. Right Triangle Trigonometry is the first-quadrant restriction of these definitions. Complex Numbers embed the circle in the plane $C$ , turning $(cos θ, sin θ)$ into $e^{i θ}$ via Euler's Formula. The smooth wobble of sine motivates Derivatives of Trigonometric Functions in calculus, and decomposing arbitrary periodic signals into sines and cosines is the heart of Fourier Series. The rotation matrix in [9] links trigonometry to Linear Algebra and Rotations.

Real-World Applications

Audio engineering. A pure musical tone is a sine wave; chords and timbres are sums of sines (Fourier synthesis) used in synthesizers and MP3 compression.
Alternating current (AC) power. Household voltage in many countries is $V (t) = V_{0} sin (2 π \cdot 50 t)$ — a 50 Hz sinusoid; impedance analysis uses $sin / cos$ phase relations.
GPS and surveying. Triangulation computes distances and positions from measured angles using the law of sines and cosines.
Computer graphics and games. Rotating sprites and 3-D models uses the rotation matrix [9]; circular and orbital motion uses $(cos θ, sin θ)$ directly.
Astronomy. Predicting planetary positions, eclipse timing, and tides relies on sums of sinusoids — the original motivation of ancient trigonometry.
Signal processing & telecommunications. Amplitude- and frequency-modulated radio signals are encoded as $cos (2 π f t)$ carriers.
Mechanical engineering. The motion of pistons, pendulums, and springs (simple harmonic motion) is described by $x (t) = A cos (ω t + φ)$ .

Historical Development

~150 BCE — Hipparchus of Nicaea. Compiles the first known table of chords, the ancestor of the sine table; often called the founder of trigonometry.
~140 CE — Ptolemy. His Almagest gives a refined chord table and the equivalent of addition formulas ("Ptolemy's theorem").
~500 CE — Aryabhata. Introduces the half-chord (jya), essentially the modern sine, and tabulates it in his Aryabhatiya.
~628 CE — Brahmagupta; ~830 CE — al-Khwarizmi and successors. Indian and Islamic astronomers develop sine, cosine, and versine tables to high precision.
~990 CE — Abu al-Wafa. Establishes all six trigonometric functions and proves the addition and half-angle formulas systematically.
1464 — Regiomontanus. De triangulis omnimodis presents trigonometry as an independent European discipline.
1593–1614 — Viète, Napier. Analytic identities and logarithms make trigonometric computation practical for navigation.
1748 — Euler. Introductio defines $sin, cos$ as numerical functions, proves $e^{i θ} = cos θ + i sin θ$ , unifying the subject with analysis.
1822 — Fourier. Shows that essentially any periodic function decomposes into sines and cosines, vastly expanding the functions' reach.

Subtleties & Limits

The geometric unit-circle definition is intuitive but not logically self-contained: it presupposes that "arc length" and "rotation by an angle" are well-defined, which in turn rests on the analytic theory of arc length and the existence of $π$ . Rigorous mathematics therefore often defines $sin$ and $cos$ by their power series $sin x = \sum (- 1)^{n} x^{2 n + 1} / (2 n + 1)!$ and proves periodicity and the geometric meaning as theorems. A genuinely subtle question is the arithmetic nature of trig values: while $cos \frac{π}{5}$ is algebraic, the values of $sin$ , $cos$ , $tan$ at rational multiples of $π$ versus at rational degrees are governed by Niven's theorem (the only rational values of $sin θ$ at rational $θ^{\circ}$ are $0, \pm \frac{1}{2}, \pm 1$ ) — and transcendence of values like $cos 1$ (radian) connects to deep open territory in number theory.