Unit-circle trigonometry

Intuition [Beginner]

Right-triangle trigonometry defines sine and cosine for angles between $0$ and $90$ degrees. But angles can be larger than $90$ degrees, or negative, or even go around the circle multiple times. The unit circle extends the definitions to all angles.

Draw a circle of radius $1$ centred at the origin. Start at the point $(1,0)$ on the right. Walk counter-clockwise around the circle by an angle $\theta$. The point where you land has coordinates $(\cos \theta ,\sin \theta )$. The $x$-coordinate is the cosine, and the $y$-coordinate is the sine.

This picture works for every angle. If $\theta =180$ degrees, you are at $(-1,0)$, so $\cos 180^{\circ }=-1$ and $\sin 180^{\circ }=0$. If $\theta =270$ degrees, you are at $(0,-1)$, so $\cos 270^{\circ }=0$ and $\sin 270^{\circ }=-1$. The circle gives coordinates, not just ratios.

Visual [Beginner]

The unit circle in the coordinate plane, with four key points marked: $(1,0)$ at $0^{\circ }$, $(0,1)$ at $90^{\circ }$, $(-1,0)$ at $180^{\circ }$, and $(0,-1)$ at $270^{\circ }$. A point $P$ at angle $\theta$ from the positive $x$-axis has coordinates $(\cos \theta ,\sin \theta )$.

In each quadrant, the signs of sine and cosine follow a pattern: both positive in quadrant I, sine positive in II, both negative in III, cosine positive in IV. The circle keeps track of the signs automatically.

Worked example [Beginner]

Find $\cos 150^{\circ }$ and $\sin 150^{\circ }$ using the unit circle.

Step 1. The angle $150^{\circ }$ is in quadrant II (between $90^{\circ }$ and $180^{\circ }$). The reference angle is $180^{\circ }-150^{\circ }=30^{\circ }$.

Step 2. In quadrant II, cosine is negative and sine is positive. The reference angle gives the magnitudes: $|\cos 150^{\circ }|=\cos 30^{\circ }=\sqrt{3}/2$ and $|\sin 150^{\circ }|=\sin 30^{\circ }=1/2$.

Step 3. Apply signs: $\cos 150^{\circ }=-\sqrt{3}/2$ and $\sin 150^{\circ }=1/2$.

What this tells us: the unit circle extends the special-angle values from 00.06.01 to all four quadrants. The reference angle gives the magnitude, and the quadrant gives the sign.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $\theta$ be a real number. The unit-circle definitions of sine and cosine are:

Definition. Let $P$ be the point on the unit circle $x^2+y^2=1$ obtained by rotating the point $(1,0)$ counter-clockwise by $\theta$ radians around the origin. Then:

$$\cos \theta =x_P,\sin \theta =y_P.$$

The tangent is defined by $\tan \theta =\sin \theta /\cos \theta$ whenever $\cos \theta \ne 0$.

Radian measure. One radian is the angle that subtends an arc of length $1$ on the unit circle. A full revolution is $2\pi$ radians ($\approx 6.283$), so $360^{\circ }=2\pi$ rad. The conversion is:

$${\theta }_{\text{rad}}={\theta }_{\text{deg}}\cdot \frac{\pi }{180},{\theta }_{\text{deg}}={\theta }_{\text{rad}}\cdot \frac{180}{\pi }.$$

Radian measure is the natural unit because on the unit circle, the arc length equals the angle in radians. This makes derivatives of trigonometric functions take their simplest form: $(d/d\theta )\sin \theta =\cos \theta$ only in radians.

Counterexamples to common slips

• $\sin \theta$ can be negative. The right-triangle definition produces only non-negative values (sides of a triangle are positive). The unit-circle definition extends $\sin \theta$ to all real numbers, including negatives in quadrants III and IV.
• $\cos \theta$ and $\sin \theta$ are not ratios of sides for obtuse angles. The unit-circle definition replaces side ratios with coordinates. The right-triangle ratio is a special case (quadrant I only).
• The period of sine and cosine is $2\pi$, not $\pi$. Both functions repeat after one full revolution: $\sin (\theta +2\pi )=\sin \theta$. The tangent function has period $\pi$: $\tan (\theta +\pi )=\tan \theta$.

Key theorem with proof [Intermediate+]

Theorem (Pythagorean identity for all real angles). For every real number $\theta$:

$${\sin }^2\theta +{\cos }^2\theta =1.$$

Proof. By the unit-circle definition, the point $P=(\cos \theta ,\sin \theta )$ lies on the circle $x^2+y^2=1$. Substituting $x_P=\cos \theta$ and $y_P=\sin \theta$ into the equation of the circle gives ${\cos }^2\theta +{\sin }^2\theta =1$. This holds for every $\theta$ because $P$ lies on the unit circle by definition, regardless of which quadrant $\theta$ places it in. $\square$

Corollary (parity relations). For all real $\theta$: $\sin (-\theta )=-\sin \theta$ (odd function) and $\cos (-\theta )=\cos \theta$ (even function).

Proof of parity. The point at angle $-\theta$ is the reflection of the point at angle $\theta$ across the $x$-axis. This reflection flips the $y$-coordinate and preserves the $x$-coordinate: $(\cos \theta ,\sin \theta )\to (\cos \theta ,-\sin \theta )$. Therefore $\cos (-\theta )=\cos \theta$ and $\sin (-\theta )=-\sin \theta$. $\square$

Bridge. The foundational reason the Pythagorean identity holds for all angles is that the unit-circle definition places every point on the circle $x^2+y^2=1$ by construction. This is exactly the content that identifies the trigonometric functions with the coordinate functions of the circle, and the bridge is between the right-triangle Pythagorean identity (a property of acute angles) and the global identity (a property of the circle's equation). The central insight is that the unit-circle definition unifies all four quadrants under a single algebraic relation, and putting these together, the identity ${\sin }^2\theta +{\cos }^2\theta =1$ generalises to the family $1+{\tan }^2\theta ={\sec }^2\theta$ and ${\cot }^2\theta +1={\csc }^2\theta$. This pattern builds toward 00.05.03 where the Euler identity $e^{i\theta }=\cos \theta +i\sin \theta$ encodes the Pythagorean identity as $|e^{i\theta }{|}^2=1$, and appears again in 00.08.01 where the addition formulas produce the full system of trigonometric identities.

Exercises [Intermediate+]

Advanced results [Master]

Theorem 1 (parametric description of the circle). The unit circle is the image of the map $\theta \mapsto (\cos \theta ,\sin \theta )$ from $\mathbb{R}$ to ${\mathbb{R}}^2$. This map is periodic with period $2\pi$ and restricts to a bijection $[0,2\pi )\to S^1$. The angular velocity is $d\theta /dt=1$ rad/s for uniform motion at unit speed on the circle.

Theorem 2 (periodicity). The functions $\sin$ and $\cos$ are periodic with minimal period $2\pi$: $\sin (\theta +2\pi )=\sin \theta$ and $\cos (\theta +2\pi )=\cos \theta$ for all $\theta$. The function $\tan$ is periodic with minimal period $\pi$. No smaller positive period exists for any of these functions.

Theorem 3 (parity). $\sin$ is an odd function: $\sin (-\theta )=-\sin \theta$. $\cos$ is an even function: $\cos (-\theta )=\cos \theta$. These follow from the reflection symmetry of the circle across the $x$-axis.

Theorem 4 (Euler's formula and the complex exponential). The identity $e^{i\theta }=\cos \theta +i\sin \theta$ (proved from power series in 00.05.03) identifies the point $(\cos \theta ,\sin \theta )$ on the unit circle with the complex number $e^{i\theta }$ of modulus $1$. The Pythagorean identity becomes $|e^{i\theta }{|}^2=1$. Multiplication by $e^{i\theta }$ is a rotation of the complex plane by angle $\theta$.

Theorem 5 (arc-length parameterisation). On the unit circle, the arc length from $(1,0)$ to $(\cos \theta ,\sin \theta )$ measured counter-clockwise is exactly $\theta$ (in radians). This is the foundational reason radian measure is natural: the parameter equals the arc length.

Theorem 6 (Fourier series preview). Every sufficiently regular periodic function of period $2\pi$ can be expressed as a (possibly infinite) sum of sines and cosines:

$$f(\theta )=\frac{a_0}{2}+\sum _{n=1}^{\infty }\left(a_n\cos (n\theta )+b_n\sin (n\theta )\right).$$

The functions $\cos (n\theta )$ and $\sin (n\theta )$ are the harmonics of the fundamental frequency $1/(2\pi )$. This decomposition is the foundation of signal processing, heat conduction, and quantum mechanics.

Theorem 7 (Winding number and covering maps). The map $p:\mathbb{R}\to S^1$, $p(\theta )=e^{i\theta }$, is the universal covering map of the circle. Each point of $S^1$ has preimage $\theta +2\pi \mathbb{Z}$ (a coset of $2\pi \mathbb{Z}$ in $\mathbb{R}$). The winding number of a loop in $S^1$ counts how many times the loop wraps around the circle, and is the prototype for the fundamental group ${\pi }_1(S^1)\cong \mathbb{Z}$.

Synthesis. The foundational reason the unit circle is the correct setting for trigonometry is that the parameterisation $\theta \mapsto (\cos \theta ,\sin \theta )$ identifies the real line (modulo $2\pi$) with the circle, and the central insight is that the Pythagorean identity ${\sin }^2\theta +{\cos }^2\theta =1$ is the equation of the circle in disguise. This is exactly the content that identifies the trigonometric functions with the coordinate projections of a uniform circular motion, and the bridge is between the geometry of the circle and the algebra of the exponential function via Euler's formula $e^{i\theta }=\cos \theta +i\sin \theta$. Putting these together, the periodicity of $\sin$ and $\cos$ is the periodicity of the parameterisation, the parity relations are the reflection symmetries of the circle, and the Fourier decomposition represents arbitrary periodic functions as superpositions of circular motions. The pattern recurs throughout analysis: the winding number builds toward 00.05.03 where the complex exponential maps the real line onto the circle, and the trigonometric addition formulas in 00.08.01 are the algebraic expression of the fact that composition of rotations corresponds to addition of angles.

Full proof set [Master]

Proposition 1 (Pythagorean identity for all real $\theta$). ${\sin }^2\theta +{\cos }^2\theta =1$ for all $\theta \in \mathbb{R}$.

Proof. By definition, the point $P=(\cos \theta ,\sin \theta )$ lies on the unit circle $x^2+y^2=1$. Substituting $x=\cos \theta$ and $y=\sin \theta$ yields the identity. $\square$

Proposition 2 (periodicity of sine and cosine). $\sin (\theta +2\pi )=\sin \theta$ and $\cos (\theta +2\pi )=\cos \theta$ for all $\theta$. No smaller positive period exists.

Proof. Adding $2\pi$ to the angle corresponds to one full counter-clockwise revolution on the unit circle, returning to the same point. Hence $\sin (\theta +2\pi )=\sin \theta$ and $\cos (\theta +2\pi )=\cos \theta$. To see that $2\pi$ is the minimal period, note that $\cos \theta =1$ only when $\theta =2\pi k$ for integer $k$ (the point $(1,0)$ is reached only at multiples of $2\pi$). So any period $T$ must satisfy $T=2\pi k$, and the smallest positive such $T$ is $2\pi$. $\square$

Proposition 3 ($\sin (\theta +\pi )=-\sin \theta$, $\cos (\theta +\pi )=-\cos \theta$).

Proof. Adding $\pi$ to the angle rotates the point $P=(\cos \theta ,\sin \theta )$ by a half-turn, producing the antipodal point $(-\cos \theta ,-\sin \theta )$. The coordinates of this point are $(\cos (\theta +\pi ),\sin (\theta +\pi ))$ by definition, so $\cos (\theta +\pi )=-\cos \theta$ and $\sin (\theta +\pi )=-\sin \theta$. $\square$

Connections [Master]

• Complex numbers and Euler's identity 00.05.03. The unit-circle definition of $\sin \theta$ and $\cos \theta$ is the real-number precursor to Euler's formula $e^{i\theta }=\cos \theta +i\sin \theta$. The point on the unit circle at angle $\theta$ is the complex number $e^{i\theta }$, and the Pythagorean identity is the statement $|e^{i\theta }|=1$. Every property of the trigonometric functions on the circle (periodicity, parity, addition formulas) is a consequence of the exponential law $e^{i(\theta +\varphi )}=e^{i\theta }{e}^{i\varphi }$.

• Right-triangle trigonometry 00.06.01. The unit-circle definition reduces to the right-triangle definition when $0<\theta <\pi /2$. In quadrant I, the point $(\cos \theta ,\sin \theta )$ on the unit circle is the top vertex of a right triangle with hypotenuse $1$, adjacent side $\cos \theta$ (the $x$-coordinate), and opposite side $\sin \theta$ (the $y$-coordinate). The ratios from 00.06.01 are recovered as $\sin \theta =\text{opposite}/\text{hypotenuse}$ and $\cos \theta =\text{adjacent}/\text{hypotenuse}$.

• Trigonometric identities (addition formulas) 00.08.01. The addition formulas $\sin (\theta +\varphi )=\sin \theta \cos \varphi +\cos \theta \sin \varphi$ and $\cos (\theta +\varphi )=\cos \theta \cos \varphi -\sin \theta \sin \varphi$ are the algebraic expression of the geometric fact that composing two rotations by angles $\theta$ and $\varphi$ produces a rotation by $\theta +\varphi$. The unit circle provides the geometric setting, and 00.08.01 derives the algebraic identities.

Historical & philosophical context [Master]

Euler 1748 Introductio in analysin infinitorum ^[Euler1748] defined the trigonometric functions as functions of a real variable on the unit circle, severing their dependence on triangles and establishing them as analytic objects. Euler introduced radian measure (implicitly, by treating angles as arc lengths on the unit circle), gave the functions their modern notation ($\sin$, $\cos$, $\tan$), and connected them to the exponential function via $e^{i\theta }=\cos \theta +i\sin \theta$. The earlier right-triangle definitions (Ptolemy ~150, Aryabhata 476, Regiomontanus 1464) were restricted to acute angles. The extension to all real angles through the unit circle is Euler's contribution, and it opened the path to the analytic theory: power series, derivatives, differential equations, and Fourier analysis. Fourier 1822 Theorie analytique de la chaleur ^{[Fourier1822]} showed that every periodic function decomposes into sines and cosines, completing the arc from geometry through algebra to analysis.

Bibliography [Master]

@book{Euler1748,
  author = {Euler, Leonhard},
  title = {Introductio in analysin infinitorum},
  publisher = {Marcus-Michaelis Bousquet},
  address = {Lausanne},
  year = {1748}
}

@book{Fourier1822,
  author = {Fourier, Joseph},
  title = {Th\'eorie analytique de la chaleur},
  publisher = {Firmin-Didot},
  address = {Paris},
  year = {1822}
}

@book{Rudin1976,
  author = {Rudin, Walter},
  title = {Principles of Mathematical Analysis},
  publisher = {McGraw-Hill},
  edition = {3rd},
  year = {1976}
}

@book{Lang1988,
  author = {Lang, Serge},
  title = {Basic Mathematics},
  publisher = {Springer},
  year = {1988}
}