The Unit Circle & Trig Identities

1. Introduction

The unit circle is the single most important picture in trigonometry. It is just the circle of radius $1$ centered at the origin — equation $x^2 + y^2 = 1$ — yet from it flows the definition of every trigonometric function, the exact values you memorize, the signs in each quadrant, the graphs of sine and cosine, and the fundamental identities. Once you truly understand the unit circle, trigonometry stops being a pile of formulas to memorize and becomes one coherent picture.

The core idea: take an angle $\theta$ measured counterclockwise from the positive $x$-axis. Its terminal side (the rotating ray) crosses the unit circle at a point whose coordinates are exactly $(\cos\theta, \sin\theta)$. That's it — cosine is the $x$-coordinate, sine is the $y$-coordinate. Everything else is a consequence. Because the point lies on the circle, $x^2 + y^2 = 1$ instantly becomes the Pythagorean identity $\cos^2\theta + \sin^2\theta = 1$.

This article builds trigonometry from the unit circle outward: radian measure and why we use it, the six functions and their signs, reference angles, the special-angle values, the web of identities (Pythagorean, even/odd, cofunction, sum and double-angle), and techniques for simplifying expressions and solving equations.

Think of the unit circle as a number line wrapped around the origin. Each real angle $\theta$ (in radians) corresponds to exactly one point on the circle, and that point's coordinates are the cosine and sine. This viewpoint unifies right-triangle trigonometry (from geometry class) with circular trigonometry (from precalculus) into a single definition. The identities you learn are not arbitrary — they are algebraic consequences of the circle equation $x^2 + y^2 = 1$.

2. Core Concepts

2.1 Angles, Standard Position, and Radian Measure

An angle is in standard position when its vertex is at the origin and its initial side lies along the positive $x$-axis. Rotating counterclockwise gives a positive angle; clockwise gives a negative one. Angles larger than one full turn are coterminal with smaller ones (they land in the same place): $\theta$ and $\theta + 360^\circ$ share a terminal side.

We can measure rotation in degrees (a full circle is $360^\circ$) or radians. One radian is the angle that subtends an arc equal in length to the radius. Since the full circumference is $2\pi r$, a full circle is $2\pi$ radians. The master conversion is: $180^\circ = \pi \text{ radians}.$ Radians are preferred in higher math because they make formulas clean: arc length is simply $s = r\theta$ and many calculus results only hold in radians.

2.2 Defining the Six Trig Functions

For the point $(\cos\theta, \sin\theta)$ on the unit circle:

• $\cos\theta = x$ (the horizontal coordinate),
• $\sin\theta = y$ (the vertical coordinate),
• $\tan\theta = \dfrac{\sin\theta}{\cos\theta} = \dfrac{y}{x}$ (the slope of the terminal side).

The three reciprocals are:

• $\csc\theta = \dfrac{1}{\sin\theta}$, $\quad\sec\theta = \dfrac{1}{\cos\theta}$, $\quad\cot\theta = \dfrac{\cos\theta}{\sin\theta}$.

This matches the right-triangle definitions ("SOH-CAH-TOA": sine $=$ opposite/hypotenuse, etc.) because in the unit circle the hypotenuse is $1$, so opposite/hypotenuse $= y/1 = y = \sin\theta$.

2.3 Signs by Quadrant

The signs of $x$ and $y$ change by quadrant, so the trig functions do too. A handy mnemonic is "All Students Take Calculus" for which functions are positive:

• Quadrant I ($0$ to $90^\circ$): All positive.
• Quadrant II ($90^\circ$ to $180^\circ$): Sine (and csc) positive.
• Quadrant III ($180^\circ$ to $270^\circ$): Tangent (and cot) positive.
• Quadrant IV ($270^\circ$ to $360^\circ$): Cosine (and sec) positive.

2.4 Reference Angles

A reference angle is the acute angle between the terminal side and the $x$-axis. Every angle's trig values equal those of its reference angle, up to a sign determined by the quadrant. To find it: in QI it's $\theta$ itself; in QII use $180^\circ - \theta$; in QIII use $\theta - 180^\circ$; in QIV use $360^\circ - \theta$ (use $\pi$ in place of $180^\circ$ for radians). This is why you only need to memorize the first-quadrant special values.

2.5 The Special Angles

From the $45$-$45$-$90$ and $30$-$60$-$90$ triangles, the first-quadrant exact values are:

• $\theta = 0$: $(\cos, \sin) = (1, 0)$.
• $\theta = \tfrac{\pi}{6}\,(30^\circ)$: $\left(\tfrac{\sqrt 3}{2}, \tfrac{1}{2}\right)$.
• $\theta = \tfrac{\pi}{4}\,(45^\circ)$: $\left(\tfrac{\sqrt 2}{2}, \tfrac{\sqrt 2}{2}\right)$.
• $\theta = \tfrac{\pi}{3}\,(60^\circ)$: $\left(\tfrac{1}{2}, \tfrac{\sqrt 3}{2}\right)$.
• $\theta = \tfrac{\pi}{2}\,(90^\circ)$: $(0, 1)$.

A memory trick: write the sines of $0, 30, 45, 60, 90^\circ$ as $\tfrac{\sqrt 0}{2}, \tfrac{\sqrt 1}{2}, \tfrac{\sqrt 2}{2}, \tfrac{\sqrt 3}{2}, \tfrac{\sqrt 4}{2}$. Cosine is the same list reversed.

2.6 The Fundamental Identities

The most important identity comes straight from $x^2 + y^2 = 1$: $\sin^2\theta + \cos^2\theta = 1.$ Dividing this equation by $\cos^2\theta$ gives $\tan^2\theta + 1 = \sec^2\theta$; dividing by $\sin^2\theta$ gives $1 + \cot^2\theta = \csc^2\theta$. These three are the Pythagorean identities.

Because reflecting the angle across the $x$-axis ($\theta \to -\theta$) keeps $x$ but negates $y$, we get the even/odd identities: $\cos(-\theta) = \cos\theta$ (even), $\sin(-\theta) = -\sin\theta$ (odd), and $\tan(-\theta) = -\tan\theta$ (odd).

The cofunction identities relate sine and cosine of complementary angles: $\cos\theta = \sin\!\left(\tfrac{\pi}{2} - \theta\right)$ and $\sin\theta = \cos\!\left(\tfrac{\pi}{2} - \theta\right)$ — the reason "co"sine is named for the complement.

2.7 Sum, Difference, and Double-Angle Formulas

For combining angles: $\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B,$ $\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B.$ Setting $A = B = \theta$ gives the double-angle formulas: $\sin 2\theta = 2\sin\theta\cos\theta, \qquad \cos 2\theta = \cos^2\theta - \sin^2\theta = 1 - 2\sin^2\theta = 2\cos^2\theta - 1.$ These let you rewrite combined or doubled angles in terms of single-angle values.

2.8 Period and Coterminal Angles

Sine and cosine repeat every full rotation: $\sin(\theta + 2\pi) = \sin\theta$ and $\cos(\theta + 2\pi) = \cos\theta$. The period is $2\pi$ radians ($360^\circ$). Tangent repeats every $\pi$ radians because $\tan(\theta + \pi) = \tan\theta$. When solving equations, add $2\pi n$ (or $360^\circ n$) to capture all solutions.

2.9 Reciprocal and Quotient Identities

Beyond the definitions: $\tan\theta = \frac{\sin\theta}{\cos\theta}, \quad \cot\theta = \frac{\cos\theta}{\sin\theta}, \quad \sec\theta = \frac{1}{\cos\theta}, \quad \csc\theta = \frac{1}{\sin\theta}.$ These let you rewrite any expression in terms of sines and cosines only — the standard move for simplification and proof.

2.10 Domain Restrictions

$\tan\theta$, $\sec\theta$, $\csc\theta$, and $\cot\theta$ are undefined wherever their denominators vanish. Tangent and secant fail at $\cos\theta = 0$ ($\theta = \tfrac{\pi}{2} + n\pi$); cosecant and cotangent fail at $\sin\theta = 0$ ($\theta = n\pi$). State exclusions when simplifying rational trig expressions.

2.11 Arc Length on the Unit Circle

On the unit circle ($r = 1$), arc length equals the radian measure of the angle: $s = \theta$. This is why radians are defined the way they are — the angle is the arc length on the unit circle.

2.12 Half-Angle Formulas (Preview)

$\sin^2\!\left(\frac{\theta}{2}\right) = \frac{1 - \cos\theta}{2}, \qquad \cos^2\!\left(\frac{\theta}{2}\right) = \frac{1 + \cos\theta}{2}.$

2.13 The Unit Circle and Trig Graphs (Preview)

As $\theta$ increases, the point $(\cos\theta, \sin\theta)$ traces the unit circle. The $y$-coordinate $\sin\theta$ oscillates between $-1$ and $1$ with period $2\pi$; the $x$-coordinate $\cos\theta$ does the same, shifted. Understanding the circle explains why sine and cosine graphs are waves, why they have the same shape, and why $\cos\theta = \sin\!\left(\theta + \tfrac{\pi}{2}\right)$ — a quarter-turn along the circle shifts sine to cosine. These follow from the double-angle formulas and help integrate powers of sine and cosine in calculus, and simplify expressions like $\sqrt{\tfrac{1-\cos\theta}{2}}$.