Unit Circle & Trigonometric Functions

Explore the 6 trigonometric ratios as geometric segments and graph the general sinusoid with period, frequency and phase shift

Angles:

Angle θ

Angle in standard position: measured from the positive x-axis, counterclockwise. Negative values and angles beyond one turn are supported.

0360

Notable angles

Position on the circle

QuadrantI

Reference angle60°

Coterminal (±1 turn)-300° · 420°

Signs

sin +cos +tan +

Exact values · θ = 60°

\operatorname{sin}\,\theta = \frac{\sqrt{3}}{2}

≈ 0.866

\operatorname{cos}\,\theta = \frac{1}{2}

≈ 0.5

\operatorname{tan}\,\theta = \sqrt{3}

≈ 1.732

\operatorname{cot}\,\theta = \frac{\sqrt{3}}{3}

≈ 0.5774

\operatorname{sec}\,\theta = 2

\operatorname{csc}\,\theta = \frac{2\sqrt{3}}{3}

≈ 1.155

Results

sin θ

0.866

cos θ

0.5

tan θ

1.732

Reciprocal functions

csc θ

1.155

sec θ

cot θ

0.5774

Fundamentals & Explanation

Unit-Circle Definition

P = (\cos\theta,\ \sin\theta)

\tan\theta = \frac{\sin\theta}{\cos\theta} \qquad \cot\theta = \frac{\cos\theta}{\sin\theta}

\sec\theta = \frac{1}{\cos\theta} \qquad \csc\theta = \frac{1}{\sin\theta}

Pythagorean Identity

\sin^2\theta + \cos^2\theta = 1

This is the Pythagorean Theorem applied to the triangle with legs $\cos\theta$ and $\sin\theta$ and hypotenuse 1 (the radius). Dividing by $\cos^2\theta$ or $\sin^2\theta$ yields $1+\tan^2\theta=\sec^2\theta$ and $1+\cot^2\theta=\csc^2\theta$ .

Reference Angle & Coterminals

\theta_{ref} \in [0°, 90°] \quad\text{(acute angle to the } x\text{-axis)}

\theta \sim \theta + 360°k, \quad k \in \mathbb{Z}

Coterminal angles share the same terminal side, and therefore all six ratios.

Notable Exact Values

θ	sin	cos	tan
0°	0	1	0
30°	$\tfrac{1}{2}$	$\tfrac{\sqrt{3}}{2}$	$\tfrac{\sqrt{3}}{3}$
45°	$\tfrac{\sqrt{2}}{2}$	$\tfrac{\sqrt{2}}{2}$	1
60°	$\tfrac{\sqrt{3}}{2}$	$\tfrac{1}{2}$	$\sqrt{3}$
90°	1	0	—

The remaining notable angles (120°, 135°, … 330°) follow from the reference angle plus the quadrant sign.

1. Why the unit circle generalizes the triangle

In a right triangle, sine and cosine are ratios of sides, which only makes sense for acute angles (0° to 90°). The circle of radius 1 removes that restriction: placing the angle $\theta$ in standard position, its terminal side meets the circle at the point $P=(\cos\theta, \sin\theta)$ . The ratios become coordinates, defined for any angle — obtuse, negative, or spanning several turns — each with its own sign. This is the definition used throughout calculus and physics.

2. Quadrant signs (ASTC) and the reference angle

The sign of each function depends only on the quadrant of the terminal side. The ASTC mnemonic (All, Sine, Tangent, Cosine) walks quadrants I→IV: in Q1 all are positive; in Q2 only sine (and cosecant); in Q3 only tangent (and cotangent); in Q4 only cosine (and secant). The magnitude, on the other hand, depends only on the reference angle — the acute angle between the terminal side and the x-axis. That is why $\sin 150° = +\sin 30° = \tfrac{1}{2}$ and $\cos 150° = -\cos 30° = -\tfrac{\sqrt{3}}{2}$ : reference 30°, quadrant-II signs.

3. The six ratios as segments

Each trigonometric ratio is the (signed) length of a concrete segment in the diagram: cosine lies along the x-axis and sine is the vertical rise up to $P$ . Tangent lives on the line $x=1$ — literally the geometric tangent to the circle, hence its name: it is the stretch from (1, 0) to where the (extended) terminal side crosses that line. Cotangent does the same on the line $y=1$ . Secant and cosecant are the distances from the origin to those crossings, measured along the terminal side (secant = “the one that cuts”). When the terminal side becomes parallel to a tangent line, the crossing no longer exists: the function is undefined and an asymptote appears — tan and sec at 90° and 270°; cot and csc at 0° and 180°.

4. From the circle to the wave

If the point $P$ spins at constant speed and we plot its height $\sin\theta$ against the angle swept, the circle “unrolls” into the sinusoid — the wave strip under the diagram shows exactly that. One full turn (360° or $2\pi$ ) is one period: this is why trigonometric functions are periodic and coterminal angles repeat their values. The general form $y = A\sin(B(x-C))+D$ merely rescales that unrolling: vertical stretch factor $|A|$ (amplitude for sin/cos), period $2\pi/|B|$ (where $B$ is horizontal compression), phase shift $C$ and midline $D$ .

5. Where this shows up in science

Uniform rotation projected onto an axis is the mathematical model of every oscillatory phenomenon:

Simple harmonic motion: the position of a mass on a spring is the shadow of a rotating point.
Alternating current: voltage and current are phase-shifted sinusoids.
Complex numbers and phasors: Euler's formula $e^{i\theta}=\cos\theta + i\sin\theta$ is the unit circle in the complex plane.
Taylor series: in practice, sin and cos are computed from their polynomial expansions.
Optics (Snell's law): refraction is governed by ratios of sines.

Sibling module & references

To solve triangles (laws of sines and cosines, SSS–SSA cases) use Analytical Trigonometry. References: Stewart, Precalculus (Trigonometric Functions); Wolfram MathWorld, Unit Circle and Trigonometric Functions; Khan Academy, Trigonometry.

Unit Circle & Trigonometric Functions

Position on the circle

Exact values · θ = 60°

Results

Reciprocal functions

Fundamentals & Explanation

1. Why the unit circle generalizes the triangle

2. Quadrant signs (ASTC) and the reference angle

3. The six ratios as segments

4. From the circle to the wave

5. Where this shows up in science

Sibling module & references

You may also like:

1. Why the unit circle generalizes the triangle

2. Quadrant signs (ASTC) and the reference angle

3. The six ratios as segments

4. From the circle to the wave

5. Where this shows up in science

Sibling module & references