Trigonometric Functions

Trigonometric functions (sine, cosine, tangent, and their reciprocals) relate the angles of a triangle to ratios of its side lengths, and more generally express periodic oscillation via the unit circle. Key result: for any real angle

θ

, the point

(cos θ, sin θ)

lies on the unit circle, giving

sin^{2} θ + cos^{2} θ = 1

. Where it applies: geometry, waves, oscillations, navigation, signal processing, and all of physics involving rotation or periodicity. Who/when: developed from Hipparchus (~150 BCE) through Indian, Islamic, and European mathematicians into the modern functional form by Euler (18th century).

The trigonometric functions are six functions of an angle, originally defined as ratios of sides in a right triangle and later extended to all real numbers using the unit circle. They matter because they are the fundamental language of periodicity: any repetitive phenomenon — sound, light, alternating current, planetary motion — can be described in terms of sines and cosines. A surprising fact is that, although they arise from triangle geometry, sine and cosine can be defined purely by infinite power series and are intimately linked to the exponential function through Euler's formula $e^{i θ} = cos θ + i sin θ$ , fusing geometry, algebra, and analysis into a single identity.

The Core Insight

A single rotating point on a circle of radius 1 encodes both sine and cosine simultaneously: its horizontal coordinate is $cos θ$ and its vertical coordinate is $sin θ$ . As the angle sweeps round, these coordinates trace out smooth waves. Every trigonometric identity is, at heart, a statement about this rotating point and the geometry of the circle.

Formal Statement

Right-Triangle Definitions

For an acute angle $θ$ in a right triangle, label the side opposite $θ$ as $O$ , the side adjacent to $θ$ as $A$ , and the hypotenuse as $H$ (the side opposite the right angle). Then:

sin θ = \frac{O}{H}, cos θ = \frac{A}{H}, tan θ = \frac{O}{A}

⟳These are definitions; the mnemonic SOH-CAH-TOA encodes them. Ratios are independent of triangle size by similarity.

The three reciprocal functions are:

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

⟳Cosecant, secant, cotangent are defined as reciprocals of sine, cosine, tangent respectively.

Here $O, A, H > 0$ are side lengths and $0 < θ < 90°$ (or $0 < θ < π /2$ radians).

Unit-Circle Definitions

To extend the functions to all real angles, place a point $P$ on the unit circle $x^{2} + y^{2} = 1$ at angle $θ$ measured counterclockwise from the positive $x$ -axis. Then:

cos θ = x_{P}, sin θ = y_{P}

⟳For a unit circle,

H = 1

, so

sin θ = O /1 = y

and

cos θ = A /1 = x

, generalised by allowing

x, y

to be negative.

Angles are best measured in radians, where a full revolution is $2 π$ :

θ_{rad} = θ_{deg} \times \frac{π}{180}

⟳One full circle subtends arc length

2 π r

; defining radian measure as arc/radius gives

2 π

rad

= 360°

Pythagorean Identity

For every real $θ$ :

sin^{2} θ + cos^{2} θ = 1

⟳Apply the equation of the unit circle

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

with

x = cos θ

y = sin θ

Dividing by $cos^{2} θ$ and by $sin^{2} θ$ yields two companions:

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

⟳Divide [5] through by

cos^{2} θ

for the first; by

sin^{2} θ

for the second; use definitions [2].

Angle Sum Formulas

For all real $α, β$ :

sin (α \pm β) = sin α cos β \pm cos α sin β

⟳Rotate a unit vector by

α

then

β

; equate components, or use the rotation matrix product / Euler's formula.

cos (α \pm β) = cos α cos β \mp sin α sin β

⟳Same rotation argument; note the sign reversal — "

\pm

" on the left gives "

\mp

" on the right.

tan (α \pm β) = \frac{tan α \pm tan β}{1 \mp tan α tan β}

⟳Divide [7] by [8] and simplify by dividing numerator and denominator by

cos α cos β

Why It Holds / Derivation

Pythagorean Identity from the Unit Circle

Define $P = (cos θ, sin θ)$ as the point on the unit circle at angle $θ$ , by [3].
By construction $P$ lies on the circle $x^{2} + y^{2} = 1$ .
Substitute $x = cos θ$ and $y = sin θ$ directly into the circle equation.
This gives $cos^{2} θ + sin^{2} θ = 1$ , establishing [5].
Because $P$ exists for every real $θ$ (the point simply wraps around the circle), the identity holds universally — not merely for acute angles.

∎

Cosine Sum Formula via Distance

Place two points on the unit circle: $A = (cos α, sin α)$ and $B = (cos β, sin β)$ .
The chord length squared is $∣ A B ∣^{2} = (cos α - cos β)^{2} + (sin α - sin β)^{2}$ .
Expanding and using [5] twice: $∣ A B ∣^{2} = 2 - 2 (cos α cos β + sin α sin β)$ .
Now rotate both points so $B$ sits at angle $0$ and $A$ at angle $(α - β)$ . Rotation preserves distance, so $∣ A B ∣^{2} = (cos (α - β) - 1)^{2} + sin^{2} (α - β)$ .
Expanding step 4: $∣ A B ∣^{2} = 2 - 2 cos (α - β)$ .
Equate steps 3 and 5: $cos (α - β) = cos α cos β + sin α sin β$ , which is [8] with the " $-$ " case.

∎

Leonhard Euler | 1707–1783

Euler transformed trigonometry from a study of triangles into a branch of analysis. He standardised the modern notation ( $sin$ , $cos$ , $tan$ ), treated the functions as functions of a real variable rather than chords, and discovered the celebrated identity $e^{i θ} = cos θ + i sin θ$ , which unifies trigonometric and exponential functions. From this single formula, every angle-sum identity falls out as a consequence of the laws of exponents.

Building Intuition

Imagine a clock hand of length 1 pinned at the origin. As it spins counterclockwise, its tip has a shadow on the floor (the $x$ -axis) and a shadow on the wall (the $y$ -axis). The floor-shadow is $cos θ$ ; the wall-shadow is $sin θ$ . When the hand points right, the floor-shadow is full (cos = 1) and the wall-shadow vanishes (sin = 0). A quarter-turn later they swap. This is why sine and cosine are the same wave shifted by a quarter cycle: $cos θ = sin (θ + π /2)$ . The tangent measures the slope of the hand — it shoots to infinity when the hand points straight up because a vertical line has undefined slope.

The unit circle generating sine as height and cosine as horizontal projection

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Worked Examples

Finding a Side with Tangent

Given: A ladder leans against a wall at an angle of

θ = 65°

to the ground. Its base is

2.0 m

from the wall. How high up the wall does it reach? Steps:

The adjacent side (base distance) is $A = 2.0 m$ ; the opposite side (height $h$ ) is unknown.
Use $tan θ = O / A$ from [1]: $tan 65° = h /2.0$ .
$tan 65° \approx 2.1445$ .
Solve: $h = 2.0 \times 2.1445 = 4.289 m$ .

h \approx 4.3 m

Evaluating an Exact Angle-Sum

Given: Find the exact value of

sin 75°

. Steps:

Write $75° = 45° + 30°$ and apply [7] with the " $+$ " case.
$sin (45° + 30°) = sin 45° cos 30° + cos 45° sin 30°$ .
Substitute exact values: $sin 45° = cos 45° = \frac{2}{2}$ , $cos 30° = \frac{3}{2}$ , $sin 30° = \frac{1}{2}$ .
$= \frac{2}{2} \cdot \frac{3}{2} + \frac{2}{2} \cdot \frac{1}{2} = \frac{6}{4} + \frac{2}{4}$ .

sin 75° = \frac{6 + 2}{4} \approx 0.9659

Common Misconceptions

$sin^{2} θ$ does not mean $sin (sin θ)$ . The notation

sin^{2} θ

is shorthand for

(sin θ)^{2}

. However,

sin^{- 1} θ

does not mean

1/ sin θ

— it denotes the inverse function (arcsine). The reciprocal is written

csc θ

(sin θ)^{- 1}

. This inconsistency in the "

^{- 1}

" superscript is a frequent source of error. Always check whether your calculator is in degrees or radians. Computing

sin 30

in radian mode gives

\approx - 0.988

(the sine of 30 radians), not the intended

0.5

. Calculus formulas such as

\frac{d}{d θ} sin θ = cos θ

are only valid in radians. Sine is not linear:

sin (α + β) \neq = sin α + sin β

in general. Take

α = β = 90°

. Then

sin (180°) = 0

, but

sin 90° + sin 90° = 1 + 1 = 2

. The correct expansion requires the angle-sum formula [7].

Connections

Trigonometric functions sit at the crossroads of many fields. Through Euler's Formula they merge with the complex Exponential Function, so that $cos θ$ and $sin θ$ become the real and imaginary parts of $e^{i θ}$ . Their derivatives and integrals are central to Calculus, where $sin$ and $cos$ form a closed cycle under differentiation. The Law of Sines and Law of Cosines extend the right-triangle ratios to any triangle. In analysis, sums of sines and cosines build the Fourier Series, decomposing arbitrary periodic signals into pure tones.

Real-World Applications

Navigation & GPS: Spherical trigonometry computes great-circle distances between latitude/longitude points on the Earth.
Alternating current (AC): Household voltage varies as $V (t) = V_{0} sin (2 π f t)$ , with $f = 50$ or $60$ Hz.
Sound & music: A pure musical tone is a sine wave; timbre is a Fourier sum of harmonic sinusoids.
Surveying: Triangulation uses measured angles and a baseline to find inaccessible distances (e.g., mountain heights).
Computer graphics: Rotation of 2D/3D objects uses rotation matrices built from $sin$ and $cos$ .
Astronomy: Parallax — the apparent shift of stars — gives distances via $tan (angle) = baseline / distance$ .
Civil engineering: Resolving forces on inclined planes and trusses into horizontal and vertical components uses $sin$ and $cos$ .

Historical Development

~150 BCE — Hipparchus of Nicaea compiles the first known table of chords, a precursor to sine, for astronomical prediction.
~150 CE — Ptolemy refines chord tables in the Almagest and derives chord-addition formulas equivalent to the angle-sum identities.
~500 CE — Aryabhata (India) introduces the half-chord (jya), the direct ancestor of the modern sine.
~830 CE — Al-Khwārizmī & later Al-Battānī (Islamic world) develop sine and tangent tables and spherical trigonometry for astronomy and qibla direction.
~1000 CE — Abū al-Wafā states identities for the six functions and improves table accuracy.
1464 — Regiomontanus writes De triangulis omnimodis, the first European treatise treating trigonometry as an independent discipline.
1614 — John Napier invents logarithms, vastly simplifying trigonometric computation.
1748 — Leonhard Euler publishes Introductio in analysin infinitorum, defining the functions analytically and proving $e^{i θ} = cos θ + i sin θ$ .
1822 — Joseph Fourier shows that arbitrary functions can be expressed as infinite sums of sines and cosines, founding harmonic analysis.

Subtleties & Limits

At the introductory level the trig functions seem fully tamed, but subtleties remain. The inverse functions are multivalued — $sin x = 0.5$ has infinitely many solutions — so principal-value branches must be chosen carefully, a recurring difficulty in solving equations and in complex analysis. Extending the functions to complex arguments makes sine and cosine unbounded (e.g. $cos (i) \approx 1.543 > 1$ ), defying naive intuition. Deep number-theoretic questions persist too: the exact nature of values like $sin (1°)$ involves algebraic numbers, and which rational multiples of $π$ give rational sine values is settled (Niven's theorem) but related transcendence questions about sums of such values remain delicate research topics.