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.

Which quadrant does a given angle lie in?

Add or subtract full turns until the angle falls between 0° and 360°, then read off the range: 0°–90° is quadrant I, 90°–180° is II, 180°–270° is III and 270°–360° is IV. For instance 200° lies in quadrant III, and −45° is the same arm as 315°, which is quadrant IV.

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 what a negative angle or one past a full turn means.

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.

Where this is used

Real situations where you count exactly the way this lesson teaches:

The workshop
Six holes spaced evenly on a circle of radius 12 cm sit 60° apart. Rather than bending a tape along the arc, work out the coordinates: the third hole falls at 120°, that is at (12 · cos 120°, 12 · sin 120°) = (−6 cm, 10.4 cm). The minus sign says it lies 6 cm to the left of the centre, and measuring along two perpendicular directions lands it to the millimetre.
Computer graphics
Every "rotate by 15°" is this one calculation run over all the pixels. A point 100 px from the centre, turned by 210°, ends up at (100 · cos 210°, 100 · sin 210°) = (−86.6, −50). No right triangle has an angle of 210°; on the unit circle the answer simply comes out as two negative coordinates, because the point is in the third quadrant.
Stepper motors
A motor with 200 steps per revolution turns its shaft by 360° / 200 = 1.8° a step, but controllers and motion libraries take radians: 1.8° = π/100 ≈ 0.0314 rad, and a full turn is 2π ≈ 6.283 rad. Send 1.8 instead of 0.0314 and the move comes out 57 times too big — that being how many degrees there are in one radian.
Convert it here: Angle

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
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.

The other measure of an angle

An angle can be measured in something other than degrees. The same quarter circle is $90^\circ$ or $\frac{\pi}{2}$ radians — a measure in which the size of an angle is the length of the arc it cuts, rather than a convention about splitting a turn into 360 parts. That is the measure the rest of this branch uses, and the one the axes of the trigonometric graphs are scaled in; the definition, both conversions and the formulas for arc length and sector area are a topic of their own: radian measure.

Exercises

For a value of a function, type the exact value, e.g. √2/2 or −1/2. For a quadrant, give its number as a Roman numeral: I, II, III or IV.

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

cos 0° =

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.
Confusing the quadrant with the reference angle — $200^\circ$ lies in quadrant III, but its reference angle is $20^\circ$ ; two different numbers answering two different questions.
Looking for the quadrant before reducing to one turn — $430^\circ$ and $-290^\circ$ are the same arm as $70^\circ$ , which is quadrant I.
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
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