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

↗ The segment continues beyond the diagram's scale.

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

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

Fundamentals & Explanation

From the triangle to the circle

In a right triangle, sine and cosine are ratios of sides, and that only makes sense for angles between 0° and 90°. The circle of radius 1 removes the restriction: with the angle $\theta$ in standard position —measured from the positive $x$ -axis, counterclockwise— its terminal side meets the circle at a point whose coordinates are the cosine and the sine.

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

They are therefore defined for any angle, positive or negative, and beyond one full turn. The other four functions are not coordinates but quotients, and they exist only where their denominator does not vanish.

\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}

Quadrant signs and the reference angle

The sign of each function depends only on the quadrant the terminal side falls in. 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 smallest angle between the terminal side and the $x$ -axis. This holds for all six functions. That is why $\sin 150° = +\sin 30° = \tfrac{1}{2}$ and $\cos 150° = -\cos 30° = -\tfrac{\sqrt{3}}{2}$ : reference 30°, quadrant-II signs.

\theta_{ref} \in [0°,\ 90°]

On the axes the reference angle reaches those endpoints: it is 0° on the $x$ -axis and 90° on the $y$ -axis, so it is not always acute. Adding or subtracting whole turns leaves the terminal side untouched, and two coterminal angles share the value of every ratio wherever it is defined.

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

Notable exact values

At the reference angles 0°, 30°, 45°, 60° and 90° the ratios take values expressible with radicals:

Exact values of sine, cosine and tangent at the notable reference angles
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°	$1$	$0$	undefined

The remaining notable angles (120°, 135°, … 330°) come from combining this value with the quadrant sign. The reciprocals follow by inverting: where sine is 0 the cosecant does not exist, and likewise for cosine and secant.

The six ratios as segments

Each ratio is the signed length of a concrete segment in the diagram. Cosine is measured along the $x$ -axis and sine along the vertical rising to $P$ . Tangent lives on the line $x=1$ , the geometric tangent to the circle —hence the name—: it is the stretch from $(1,0)$ to where the terminal side, extended if need be, crosses that line. Cotangent does the same on the line $y=1$ .

Secant and cosecant are oriented measures from the origin to those same crossings, taken along the line of the terminal side, not along the ray: that is why they still exist when the crossing falls on the backward extension. When the terminal side becomes parallel to one of those lines the crossing disappears, the function is undefined and an asymptote appears: tangent and secant at 90° and 270°; cotangent and cosecant at 0° and 180°.

From the circle to the wave

If the point $P$ spins at constant speed and we plot its height against the angle swept, the circle “unrolls” into a wave. One full turn —360° or $2\pi$ — is one period: that is the origin of these functions being periodic. Tangent and cotangent repeat every half turn, not every turn.

y = A\,f\big(B(x-C)\big) + D

With $f$ one of the six ratios, this form merely rescales that unrolling: $|A|$ stretches vertically and is the amplitude when the base is sine or cosine; $B$ compresses the axis and sets the period to $2\pi/|B|$ for sine, cosine, secant and cosecant, and to $\pi/|B|$ for tangent and cotangent; $C$ shifts horizontally and $D$ vertically.

How to read the results

Quadrant: Where the terminal side falls, I to IV. An angle resting exactly on an axis belongs to no quadrant and is reported as such.
Reference angle: The smallest angle to the $x$ -axis, between 0° and 90°.
Coterminal: The same angle written one turn earlier and one turn later, in the chosen unit.
Undefined: The function has no value there. $\pm\infty$ is not shown, because infinity describes the behaviour on approach, not the value at the point.
Period: The length of one cycle along the $x$ -axis, in the axis unit. With $A = 0$ the graph is constant and has no period of its own.
Range: The values the function attains. Secant and cosecant leave a central gap: they never take values between $D-|A|$ and $D+|A|$ .
Asymptotes and excluded points: How many times the curve breaks inside the window. With $A = 0$ the graph is a straight line with holes, not with asymptotes.

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. Where the denominator is nonzero, dividing by $\cos^2\theta$ or $\sin^2\theta$ gives $1+\tan^2\theta=\sec^2\theta$ and $1+\cot^2\theta=\csc^2\theta$ .

Assumptions and limits

An exact value is recognised only at the notable angle itself: 60.001° and 1.0472 rad are close to 60° and to $\pi/3$ , but they are not those angles.
Quotients with no value are reported as “undefined”, never as a huge number.
The $x$ -axis of the graphs is an angle, not a time; no seconds or hertz are involved.
$B$ is dimensionless. Changing the axis unit re-expresses $C$ , the period and the window, but does not alter the curve.
The window allows up to 40 periods; beyond that the drawing stops being legible.

Solving trigonometric equations, inverse functions, symbolic identities and physical waves with units of time are all outside this scope.

Where this shows up

Simple harmonic motion: the position of a mass on a spring is the shadow of a rotating point.
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, sine and cosine are computed from their polynomial expansions.
Optics and Snell's law: refraction is governed by ratios of sines.
Solving triangles: laws of sines and cosines for the SSS and SSA cases.

References

OpenStax Precalculus 2e — 5.2 Unit Circle: Sine and Cosine Functions. Sine and cosine as coordinates, reference angle and notable values.
OpenStax Precalculus 2e — 5.3 The Other Trigonometric Functions. The four remaining ratios and where they are undefined.
OpenStax Precalculus 2e — 6.1 Graphs of the Sine and Cosine Functions. Period, amplitude and shifts. Note that it writes the form as $A\sin(Bx-C)+D$ , with phase shift $C/B$ ; here $B(x-C)$ is used, where $C$ is already the shift.
Wolfram MathWorld — Unit Circle. Definition and its connection to the complex plane.

Unit Circle & Trigonometric Functions

Results

Reciprocal functions

Position on the circle

Exact values · θ = 60°

Fundamentals & Explanation

From the triangle to the circle

Quadrant signs and the reference angle

Notable exact values

The six ratios as segments

From the circle to the wave

How to read the results

Pythagorean identity

Assumptions and limits

Where this shows up

References

From the triangle to the circle

Quadrant signs and the reference angle

Notable exact values

The six ratios as segments

From the circle to the wave

How to read the results

Pythagorean identity

Assumptions and limits

Where this shows up

References