What is the unit circle?

It is the circle centred at the origin of the coordinate system with radius 1. Every angle α, measured from the positive x semi-axis, meets it in exactly one point — and the coordinates of that point are (cos α, sin α).

What signs do the sine and the cosine take in each quadrant?

In quadrant I both are positive. In II the cosine is negative and the sine positive. In III both are negative. In IV the cosine is positive and the sine negative. Just read the coordinates of the point: the cosine carries the sign of x, the sine the sign of y.

How many radians are 180 degrees?

Exactly π radians, about 3.1416. Hence the conversion: to turn degrees into radians multiply by π/180°, and the other way round multiply by 180°/π. For example 60° = 60 · π/180 = π/3.

It is the central angle that cuts off an arc as long as the radius. Since the whole circle is 2πr long, a full turn is 2π radians. The radian is the natural measure of an angle because it rests on no convention — unlike splitting a turn into 360 parts, which we inherited from the Babylonians.

How do you find the sine of an angle greater than 90°?

On the unit circle you simply read the coordinate of the point. In practice you use the reference angle to the nearest x semi-axis and attach the sign from the quadrant: sin 150° = sin 30° = 1/2, while cos 150° = −cos 30° = −√3/2.

What does a negative angle or one greater than 360° mean?

A negative angle is measured clockwise, and an angle past 360° is a full turn with something left over. Since a full turn returns to the same point, the values repeat: sin 390° = sin 30° and sin(−30°) = −sin 30°.

Why do the sine and the cosine never exceed 1?

Because they are the coordinates of a point lying on a circle of radius 1. No coordinate of such a point can be more than 1 or less than −1, so both functions only ever take values between −1 and 1.

Intermediate level

The unit circle

Q: Why is the cosine the x-coordinate and the sine the y-coordinate?

Because for an acute angle the radius, its projection onto the x axis and the vertical segment form a right triangle whose hypotenuse is 1. Then cos α = x/1 = x and sin α = y/1 = y. A radius of one removes the division from the definition.

The unit circle carries the sine and the cosine from a triangle onto the whole circle. See how the coordinates of a point become the values of the functions, what signs they take in the four quadrants, and why the radian is the natural measure of an angle.

Before you start

This topic builds on earlier ideas. Before you start, it's worth working through the lessons below — they'll make everything click:

Sine, cosine and tangentSine, cosine and tangent are three ratios of the sides of a right triangle. See why they depend on the angle alone, where the exact values for 30°, 45° and 60° come from, and how to find a side you cannot measure.
Area and circumference of a circleThe area of a disc is A = πr², its circumference is C = 2πr. See what the number π actually is, how the radius relates to the diameter, why the answer nearly always has to be rounded, and how a circle differs from a disc.

All formulas

Equation of the unit circle
$x^2 + y^2 = 1$
a circle centred at (0, 0) with radius 1
The point on the circle
$P = (\cos \alpha, \; \sin \alpha)$
the cosine is the x-coordinate, the sine the y-coordinate
Degrees → radians
$\alpha_{\text{rad}} = \alpha_{\deg} \cdot \frac{\pi}{180^\circ}$
180° is π radians
Radians → degrees
$\alpha_{\deg} = \alpha_{\text{rad}} \cdot \frac{180^\circ}{\pi}$
the same relation the other way round
Coterminal angles
$\sin(\alpha + 360^\circ) = \sin \alpha$
a full turn changes nothing

The right-triangle definitions have one limitation: a triangle only has acute angles, so $\sin 150^\circ$ means nothing in them. The unit circle removes that limitation — and shows, along the way, what the sine and the cosine really are.

A circle of radius 1

Draw a circle centred at $(0, 0)$ with radius $1$ :

x^2 + y^2 = 1

The angle $\alpha$ is measured from the positive $x$ semi-axis, counter-clockwise. Its arm meets the circle in exactly one point $P$ .

The point P has coordinates (cos α, sin α): the cosine is the segment on the x axis, the sine the one on the y axis.

P = (\cos \alpha, \; \sin \alpha)

For an acute angle this is exactly the triangle definition. The radius, its projection onto the $x$ axis and the vertical segment form a right triangle whose hypotenuse is $1$ , so

\cos \alpha = \frac{x}{1} = x, \qquad \sin \alpha = \frac{y}{1} = y

A radius of one removes the division — only the coordinates are left. What is new is that the point can be turned past $90^\circ$ and its coordinates still exist.

Signs in the quadrants

The axes split the plane into four quadrants, numbered counter-clockwise. The sign of a function is simply the sign of the matching coordinate.

In each quadrant the first symbol belongs to the cosine (the x-coordinate), the second to the sine (the y-coordinate).

quadrant	range	$\cos \alpha$	$\sin \alpha$	$\tan \alpha$
I	$0^\circ$ – $90^\circ$	$+$	$+$	$+$
II	$90^\circ$ – $180^\circ$	$-$	$+$	$-$
III	$180^\circ$ – $270^\circ$	$-$	$-$	$+$
IV	$270^\circ$ – $360^\circ$	$+$	$-$	$-$

The tangent is a quotient, so its sign follows from the other two: two minus signs in the third quadrant make a plus.

Values outside the first quadrant

Take the numeric value from the reference angle — the acute angle between the arm and the $x$ axis — and the sign from the quadrant.

Find sin 150° and cos 150°.

Negative angles and extra turns

Nothing forces us to stop at $360^\circ$ . A negative angle is measured clockwise, and an angle past a full turn simply lands where it has already been:

\sin(\alpha + 360^\circ) = \sin \alpha, \qquad \cos(\alpha + 360^\circ) = \cos \alpha

390° is a full turn plus 30°. The arm ends where the arm of 30° ends — which is why both sines are equal.

Angles leading to the same point are called coterminal. Hence $\sin 390^\circ = \sin 30^\circ = \frac{1}{2}$ , while $\sin(-30^\circ) = -\frac{1}{2}$ , because turning the other way flips the sign of the $y$ -coordinate.

Degrees and radians

Splitting a full turn into $360$ parts is a convention — inherited from the Babylonian base-60 system and convenient because 360 divides by so many numbers. The natural measure looks different.

A radian is the central angle that cuts off an arc as long as the radius. The whole circle is $2\pi r$ long, so it holds $2\pi$ radians:

360^\circ = 2\pi \; \text{rad}, \qquad 180^\circ = \pi \; \text{rad}

which gives both conversions:

\alpha_{\text{rad}} = \alpha_{\deg} \cdot \frac{\pi}{180^\circ}, \qquad \alpha_{\deg} = \alpha_{\text{rad}} \cdot \frac{180^\circ}{\pi}

The same angle written in radians: 60° is π/3, one sixth of a half-turn.

degrees	$30^\circ$	$45^\circ$	$60^\circ$	$90^\circ$	$180^\circ$	$270^\circ$	$360^\circ$
radians	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$	$\pi$	$\frac{3\pi}{2}$	$2\pi$

Convert 135° to radians and 5π/6 to degrees.

If you need actual values converted — gradians, arcminutes or arcseconds included — our angle converter will do it.

Exercises

For a value of a function, type the exact value, e.g. √2/2 or −1/2. For a unit conversion the answer is either a multiple of π (e.g. π/3) or a plain number of degrees (e.g. 60).

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 8Score: 0

sin 90° =

Common mistakes

Measuring the angle from the $y$ axis — an angle always starts at the positive $x$ semi-axis.
Swapping sine and cosine — the cosine is the horizontal coordinate, the sine the vertical one. Alphabetically: cosine before sine, just as $x$ comes before $y$ .
Losing the sign in quadrants II–IV — the value comes from the reference angle, but the sign comes from the quadrant.
Multiplying by $180/\pi$ instead of $\pi/180$ — converting degrees to radians must make the number smaller (e.g. $60^\circ \to \frac{\pi}{3} \approx 1.05$ ).
Treating $\pi$ as a unit — $\frac{\pi}{3}$ is a number, not "π thirds of a degree".
A sine greater than 1 — impossible on a circle of radius 1; such a result means an error in the arithmetic.

Formula card

Topic: The unit circle

Equation of the unit circle
$x^2 + y^2 = 1$
a circle centred at (0, 0) with radius 1
The point on the circle
$P = (\cos \alpha, \; \sin \alpha)$
the cosine is the x-coordinate, the sine the y-coordinate
Degrees → radians
$\alpha_{\text{rad}} = \alpha_{\deg} \cdot \frac{\pi}{180^\circ}$
180° is π radians
Radians → degrees
$\alpha_{\deg} = \alpha_{\text{rad}} \cdot \frac{180^\circ}{\pi}$
the same relation the other way round
Coterminal angles
$\sin(\alpha + 360^\circ) = \sin \alpha$
a full turn changes nothing

The unit circle: the point P on the circle has coordinates (cos α, sin α). The cosine is measured along the x axis, the sine along the y axis.

Signs in the four quadrants: the first symbol is the cosine (the x-coordinate), the second the sine (the y-coordinate).

Frequently asked questions

A luminous unit circle with a radius sweeping out an angle, a sine wave unrolling from it as a streak of light, beside a right triangle drawn in stars, against a cosmic nebula in violet and magenta

Trigonometry

Why sine doesn't depend on the size of the triangle

The height of a tower, the width of a river, the distance to a ship offshore — all of it can be computed without going anywhere near the thing being measured. The whole of trigonometry rests on one observation: the ratio of two sides in a right triangle depends only on the angle, not on how big the triangle is. That is what turns an angle into a number — and lets it be tabulated once and for all.

13 min