Trigonometric Functions and the Unit Circle

⚡ Key Takeaways

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\theta measured counterclockwise from the positive xx-axis, cos⁡θ\cos\theta is the xx-coordinate and sin⁡θ\sin\theta is the yy-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.

💡 The Core Insight

A circle of radius 1 turns angle into position. Because the radius is exactly 1, the Pythagorean relation x2+y2=1x^2+y^2=1 becomes cos⁡2θ+sin⁡2θ=1\cos^2\theta+\sin^2\theta=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\theta lives in the first quadrant; the unit circle generalizes them to negative angles, obtuse angles, and angles beyond a full turn.

Formal Statement

DefinitionRadian 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 2π2\pi, so

1
360∘=2π rad,θrad=θdeg⋅π180.\quad 360^\circ = 2\pi \text{ rad}, \qquad \theta_{\text{rad}} = \theta_{\text{deg}}\cdot\frac{\pi}{180}.

Derivation hint: arc length s=rθs=r\theta with r=1r=1 gives s=θs=\theta; the whole circle has s=2πs=2\pi.

DefinitionUnit-circle trigonometric functions

Let θ∈R\theta\in\mathbb{R} be an angle measured counterclockwise (positive) from the positive xx-axis. Let P=(x,y)P=(x,y) be the point reached on the unit circle x2+y2=1x^2+y^2=1. Define

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cos⁡θ=x,sin⁡θ=y,tan⁡θ=yx=sin⁡θcos⁡θ (x≠0).\quad \cos\theta = x, \qquad \sin\theta = y, \qquad \tan\theta = \frac{y}{x}=\frac{\sin\theta}{\cos\theta}\ (x\neq 0).

Derivation hint: project PP onto each axis; tan⁡θ\tan\theta is the slope of the radius OPOP. The reciprocal functions are

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csc⁡θ=1sin⁡θ,sec⁡θ=1cos⁡θ,cot⁡θ=cos⁡θsin⁡θ,\quad \csc\theta=\frac{1}{\sin\theta},\quad \sec\theta=\frac{1}{\cos\theta},\quad \cot\theta=\frac{\cos\theta}{\sin\theta},

each defined wherever its denominator is nonzero. Derivation hint: invert sine, cosine, tangent respectively; restrict domains to avoid division by zero. Here θ\theta is the angle in radians, x=cos⁡θx=\cos\theta and y=sin⁡θy=\sin\theta are the coordinates of PP, and O=(0,0)O=(0,0) is the center.

Figure
The unit circle: the point P=(cosθ, sinθ) at angle θ
x y θ P=(cosθ, sinθ) cosθ sinθ O (1,0)
TheoremPythagorean identity

For every real θ\theta,

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cos⁡2θ+sin⁡2θ=1.\quad \cos^2\theta+\sin^2\theta = 1.

Derivation hint: P=(cos⁡θ,sin⁡θ)P=(\cos\theta,\sin\theta) lies on x2+y2=1x^2+y^2=1 by definition. Dividing by cos⁡2θ\cos^2\theta and by sin⁡2θ\sin^2\theta gives the companion forms

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1+tan⁡2θ=sec⁡2θ,1+cot⁡2θ=csc⁡2θ.\quad 1+\tan^2\theta=\sec^2\theta,\qquad 1+\cot^2\theta=\csc^2\theta.

Derivation hint: divide [4] through by cos⁡2θ\cos^2\theta (resp. sin⁡2θ\sin^2\theta) and apply [2]–[3].

TheoremPeriodicity, parity, and co-function relations

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

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sin⁡(θ+2π)=sin⁡θ,cos⁡(θ+2π)=cos⁡θ,tan⁡(θ+π)=tan⁡θ.\quad \sin(\theta+2\pi)=\sin\theta,\qquad \cos(\theta+2\pi)=\cos\theta,\qquad \tan(\theta+\pi)=\tan\theta.

Derivation hint: adding 2π2\pi returns PP to the same point; tan⁡\tan repeats every half-turn because (x,y)(x,y) and (−x,−y)(-x,-y) share a slope.

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cos⁡(−θ)=cos⁡θ (even),sin⁡(−θ)=−sin⁡θ (odd).\quad \cos(-\theta)=\cos\theta\ (\text{even}),\qquad \sin(-\theta)=-\sin\theta\ (\text{odd}).

Derivation hint: reversing rotation reflects PP across the xx-axis, flipping the sign of yy only.

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sin⁡ ⁣(π2−θ)=cos⁡θ,cos⁡ ⁣(π2−θ)=sin⁡θ.\quad \sin\!\Big(\frac{\pi}{2}-\theta\Big)=\cos\theta,\qquad \cos\!\Big(\frac{\pi}{2}-\theta\Big)=\sin\theta.

Derivation hint: reflecting θ\theta to π2−θ\tfrac{\pi}{2}-\theta swaps the xx- and yy-coordinates of PP.

Why It Holds / Derivation

ProofFrom right triangle to circle, and the addition formula

Step 1 — Right triangle case. In a right triangle with hypotenuse 1 and acute angle θ\theta, the side opposite θ\theta has length sin⁡θ\sin\theta and the adjacent side cos⁡θ\cos\theta (definitions of opposite/hypotenuse, adjacent/hypotenuse). Placing the triangle with its right angle on the xx-axis and vertex at OO puts the far vertex at exactly (cos⁡θ,sin⁡θ)(\cos\theta,\sin\theta).

Step 2 — Extension. For any real θ\theta, define PP by rotating the point (1,0)(1,0) through angle θ\theta. The coordinates of PP 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 PP lies on the circle, x2+y2=1x^2+y^2=1, i.e. equation [4].

Step 4 — Addition formula. Rotating by α\alpha then β\beta equals rotating by α+β\alpha+\beta. A 2×22\times2 rotation matrix sends (1,0)(1,0) to (cos⁡θ,sin⁡θ)(\cos\theta,\sin\theta), and composing two rotations multiplies the matrices:

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(cos⁡β−sin⁡βsin⁡βcos⁡β)(cos⁡αsin⁡α)=(cos⁡(α+β)sin⁡(α+β)).\quad \begin{pmatrix}\cos\beta&-\sin\beta\\ \sin\beta&\cos\beta\end{pmatrix}\begin{pmatrix}\cos\alpha\\ \sin\alpha\end{pmatrix}=\begin{pmatrix}\cos(\alpha+\beta)\\ \sin(\alpha+\beta)\end{pmatrix}.

Derivation hint: matrix multiply the left side and read off components. Reading off the two coordinates yields the addition formulas

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sin⁡(α+β)=sin⁡αcos⁡β+cos⁡αsin⁡β,cos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡β.\quad \sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta,\qquad \cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta.

Derivation hint: equate components in [9]. ■\blacksquare

∎
mathematician · Leonhard Euler | 1707–1783

Euler reshaped trigonometry from a table-driven craft into analysis. He standardized the notation sin⁡\sin, cos⁡\cos, tan⁡\tan as functions of a numerical (radian) argument rather than line segments in a circle, and proved the identity that bears his name, eiθ=cos⁡θ+isin⁡θe^{i\theta}=\cos\theta+i\sin\theta. 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

🧠 Intuition

Picture a clock hand of length 1 sweeping counterclockwise. Its shadow on the floor (the xx-axis) is cos⁡θ\cos\theta; its height above the floor (the yy-axis) is sin⁡θ\sin\theta. As the hand spins, the height rises and falls smoothly between +1+1 and −1-1 — 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

ExampleEvaluating at a special angle

Given: Find sin⁡θ\sin\theta, cos⁡θ\cos\theta, tan⁡θ\tan\theta for θ=5π6\theta = \dfrac{5\pi}{6} (i.e. 150∘150^\circ).

Step 1. 150∘150^\circ is in the second quadrant, where x<0x<0 and y>0y>0, so cosine is negative and sine is positive.

Step 2. Reference angle: 180∘−150∘=30∘=π6180^\circ-150^\circ=30^\circ=\tfrac{\pi}{6}. The known values are sin⁡30∘=12\sin 30^\circ=\tfrac12, cos⁡30∘=32\cos 30^\circ=\tfrac{\sqrt3}{2}.

Step 3. Apply quadrant signs:

sin⁡5π6=+12,cos⁡5π6=−32.\sin\frac{5\pi}{6}=+\tfrac12,\qquad \cos\frac{5\pi}{6}=-\tfrac{\sqrt3}{2}.

Step 4. Tangent: tan⁡θ=sin⁡θcos⁡θ=1/2−3/2=−13\tan\theta=\dfrac{\sin\theta}{\cos\theta}=\dfrac{1/2}{-\sqrt3/2}=-\dfrac{1}{\sqrt3}.

 sin⁡5π6=12,cos⁡5π6=−32,tan⁡5π6=−13≈−0.577 \boxed{\ \sin\tfrac{5\pi}{6}=\tfrac12,\quad \cos\tfrac{5\pi}{6}=-\tfrac{\sqrt3}{2},\quad \tan\tfrac{5\pi}{6}=-\tfrac{1}{\sqrt3}\approx-0.577\ }

(dimensionless ratios)

ExampleModeling a Ferris wheel

Given: A Ferris wheel of radius R=15 mR=15\text{ m} has its center 18 m18\text{ m} above the ground. It rotates once every 40 s40\text{ s}. A rider boards at the lowest point (t=0t=0). Find the rider's height h(t)h(t) and their height at t=10 st=10\text{ s}.

Step 1. Angular speed: ω=2π40=π20 rad/s\omega=\dfrac{2\pi}{40}=\dfrac{\pi}{20}\ \text{rad/s}.

Step 2. Starting at the bottom, height is lowest at t=0t=0, so use a downward-shifted cosine:

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h(t)=18−15cos⁡ ⁣(π20 t).\quad h(t)=18-15\cos\!\Big(\frac{\pi}{20}\,t\Big).

Derivation hint: center height minus vertical projection Rcos⁡(ωt)R\cos(\omega t); at t=0t=0 this gives 18−15=318-15=3 m, the lowest point.

Step 3. Evaluate at t=10 st=10\text{ s}: π20⋅10=π2\dfrac{\pi}{20}\cdot 10=\dfrac{\pi}{2}, and cos⁡π2=0\cos\tfrac{\pi}{2}=0.

h(10)=18−15⋅0=18 m.h(10)=18-15\cdot 0 = 18\text{ m}.
 h(t)=18−15cos⁡ ⁣(π20t) m,h(10)=18 m \boxed{\,h(t)=18-15\cos\!\big(\tfrac{\pi}{20}t\big)\ \text{m},\qquad h(10)=18\ \text{m}\,}

Common Misconceptions

⚠️ Common mistake

Degrees vs. radians. Calculus formulas such as ddθsin⁡θ=cos⁡θ\dfrac{d}{d\theta}\sin\theta=\cos\theta and the limit lim⁡θ→0sin⁡θθ=1\lim_{\theta\to0}\frac{\sin\theta}{\theta}=1 hold only in radians. If your calculator is in degree mode, derivatives pick up a hidden factor of π180\frac{\pi}{180}. Always check the angle unit before computing.

⚠️ Common mistake

sin⁡2θ\sin^2\theta is not sin⁡(θ2)\sin(\theta^2) and sin⁡−1\sin^{-1} is not 1/sin⁡1/\sin. The notation sin⁡2θ\sin^2\theta means (sin⁡θ)2(\sin\theta)^2, but sin⁡−1θ\sin^{-1}\theta means the inverse function arcsin⁡θ\arcsin\theta, not csc⁡θ=1/sin⁡θ\csc\theta=1/\sin\theta. These clashing conventions are a frequent source of error.

Counterexample

One might guess sin⁡(α+β)=sin⁡α+sin⁡β\sin(\alpha+\beta)=\sin\alpha+\sin\beta. Test α=β=π2\alpha=\beta=\tfrac{\pi}{2}:

sin⁡(π)=0,butsin⁡π2+sin⁡π2=1+1=2.\sin(\pi)=0,\quad\text{but}\quad \sin\tfrac{\pi}{2}+\sin\tfrac{\pi}{2}=1+1=2.

Since 0≠20\neq 2, sine is not additive; the correct rule is the addition formula [10].

Connections

connection

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\mathbb{C}, turning (cos⁡θ,sin⁡θ)(\cos\theta,\sin\theta) into eiθe^{i\theta} 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

🔬 Where this appears

Historical Development

⏳ History
- ~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⁡\sin,\cos as numerical functions, proves eiθ=cos⁡θ+isin⁡θe^{i\theta}=\cos\theta+i\sin\theta, 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

open-problem

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\pi. Rigorous mathematics therefore often defines sin⁡\sin and cos⁡\cos by their power series sin⁡x=∑(−1)nx2n+1/(2n+1)!\sin x=\sum (-1)^n x^{2n+1}/(2n+1)! and proves periodicity and the geometric meaning as theorems. A genuinely subtle question is the arithmetic nature of trig values: while cos⁡π5\cos\tfrac{\pi}{5} is algebraic, the values of sin⁡\sin, cos⁡\cos, tan⁡\tan at rational multiples of π\pi versus at rational degrees are governed by Niven's theorem (the only rational values of sin⁡θ\sin\theta at rational θ∘\theta^\circ are 0,±12,±10,\pm\tfrac12,\pm1) — and transcendence of values like cos⁡1\cos 1 (radian) connects to deep open territory in number theory.

See Also

Right Triangle Trigonometry · Complex Numbers · Euler's Formula · Graphs of Trigonometric Functions · Simple Harmonic Motion · Law of Sines and Cosines · Fourier Series