What is sin 30° on the unit circle?

sin 30° = 1/2. The 30° point on the unit circle is (√3/2, 1/2), and sine is the y-coordinate. In radians, 30° is π/6, so sin(π/6) = 1/2 as well.

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Interactive Unit Circle

Q: Why is tan 90° undefined?

Because tan θ = sin θ / cos θ, and at 90° the point on the unit circle is (0, 1), so cos 90° = 0. Dividing by zero is undefined. The same happens at 270°, where the point is (0, −1).

Q: How do you convert between degrees and radians?

A full turn is 360° or 2π radians, so 180° = π radians. To convert degrees to radians, multiply by π/180; to convert radians to degrees, multiply by 180/π. For example, 135° × π/180 = 3π/4.

The unit circle is a circle of radius 1 centered at the origin. For any angle $\theta$ , the point where it meets the circle is $(\cos \theta , \sin \theta )$ : cosine is the x-coordinate and sine is the y-coordinate. Drag the point below, or type an angle in degrees or radians, to see the exact sine, cosine and tangent, the quadrant and the reference angle. Then test yourself on the 16 standard angles. Free, and nothing is stored.

Explore the unit circle

Drag the point around the circle, use the slider, or type any angle - including negative angles and angles past a full turn.

cos θ (x) sin θ (y) angle θ

Angle θ in degrees

Snap to the standard angles when dragging

The angle

θ = 30° = π/6 rad

Position: Quadrant I - reference angle 30°
Point on the circle: (0.866, 0.5)

sin θ (the y-coordinate)1/2≈ 0.5

cos θ (the x-coordinate)√3/2≈ 0.866

tan θ = sin θ / cos θ√3/3≈ 0.5774

A standard angle, so the values are exact - the ones teachers expect on a non-calculator test.

Jump to a standard angle

Test yourself: exact values

Questions on the 16 standard angles, in degrees and radians - the exact values you need without a calculator.

Why the coordinates are cosine and sine

Drop a vertical line from the point to the x-axis and you get a right triangle whose hypotenuse is the radius, 1. By SOH-CAH-TOA, the horizontal leg is $\cos \theta \times 1$ and the vertical leg is $\sin \theta \times 1$ , so the point sits at $(\cos \theta , \sin \theta )$ . That is the whole idea, and the tool draws that triangle for you: the blue leg is cosine, the red leg is sine.

Three facts follow straight from it. Sine and cosine always lie between −1 and 1, because the circle only reaches that far. Their signs change from quadrant to quadrant, because x and y do. And because the circle has radius 1, the Pythagorean theorem gives the most important identity in trigonometry:

\displaystyle \cos ^{2} \theta + \sin ^{2} \theta = 1

Where the exact values come from

The standard angles come from two special right triangles. A 45-45-90 triangle with hypotenuse 1 has two equal legs of $\frac{\sqrt{2}}{2}$ , which gives the 45° point $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$ . A 30-60-90 triangle with hypotenuse 1 has a short leg of $\frac{1}{2}$ and a long leg of $\frac{\sqrt{3}}{2}$ , which gives the 30° point $(\frac{\sqrt{3}}{2}, \frac{1}{2})$ and, swapped, the 60° point $(\frac{1}{2}, \frac{\sqrt{3}}{2})$ .

Every other standard angle reuses those numbers. Find the reference angle (the acute angle to the x-axis: 150° and 210° both have reference angle 30°), take that angle's values, and set the signs by quadrant. Cosine is negative on the left half of the circle and sine is negative on the bottom half, which is what the memory aid "All Students Take Calculus" encodes: all positive in Quadrant I, only sine in II, only tangent in III, only cosine in IV.

Degrees and radians

Radians measure an angle by the arc it cuts off on the unit circle. The whole circumference is $2\pi$ , so a full turn is $2\pi$ radians and half a turn is $\pi$ :

\displaystyle 180^\circ = \pi \text{ rad}

Multiply by $\frac{\pi }{180}$ to go from degrees to radians and by $\frac{180}{\pi }$ to go back. Switch the tool to radians and you can type angles such as 5pi/6 or -3pi/4 directly. If your calculator gives "random" answers, check that it is in the same mode as the question: it is the most common trig mistake there is.

Unit circle chart: all 16 standard angles

Degrees	Radians	cos θ	sin θ	tan θ
0°	$0$	$1$	$0$	$0$
30°	$\frac{\pi }{6}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{3}$
45°	$\frac{\pi }{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$
60°	$\frac{\pi }{3}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\sqrt{3}$
90°	$\frac{\pi }{2}$	$0$	$1$	undefined
120°	$\frac{2\pi }{3}$	$-\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$-\sqrt{3}$
135°	$\frac{3\pi }{4}$	$-\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$-1$
150°	$\frac{5\pi }{6}$	$-\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$-\frac{\sqrt{3}}{3}$
180°	$\pi$	$-1$	$0$	$0$
210°	$\frac{7\pi }{6}$	$-\frac{\sqrt{3}}{2}$	$-\frac{1}{2}$	$\frac{\sqrt{3}}{3}$
225°	$\frac{5\pi }{4}$	$-\frac{\sqrt{2}}{2}$	$-\frac{\sqrt{2}}{2}$	$1$
240°	$\frac{4\pi }{3}$	$-\frac{1}{2}$	$-\frac{\sqrt{3}}{2}$	$\sqrt{3}$
270°	$\frac{3\pi }{2}$	$0$	$-1$	undefined
300°	$\frac{5\pi }{3}$	$\frac{1}{2}$	$-\frac{\sqrt{3}}{2}$	$-\sqrt{3}$
315°	$\frac{7\pi }{4}$	$\frac{\sqrt{2}}{2}$	$-\frac{\sqrt{2}}{2}$	$-1$
330°	$\frac{11\pi }{6}$	$\frac{\sqrt{3}}{2}$	$-\frac{1}{2}$	$-\frac{\sqrt{3}}{3}$

Each point on the circle is $(\cos \theta , \sin \theta )$ . Tangent is undefined at 90° and 270°, where $\cos \theta = 0$ .

Frequently asked questions

What is the unit circle?+

The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. Measure an angle $\theta$ counterclockwise from the positive x-axis, and the point where its terminal side meets the circle is $(\cos \theta , \sin \theta )$ . So cosine is the x-coordinate, sine is the y-coordinate, and tangent is their ratio, $\tan \theta = \sin \theta / \cos \theta$ .

How do you memorize the unit circle quickly?+

Learn how to rebuild it instead of memorizing all 16 points. In the first quadrant the sines of 0°, 30°, 45°, 60° and 90° are $\sqrt{0}/2, \sqrt{1}/2, \sqrt{2}/2, \sqrt{3}/2, \sqrt{4}/2$ , which simplify to $0, 1/2, \sqrt{2}/2, \sqrt{3}/2$ and $1$ ; the cosines are the same list in reverse. Every other standard angle has one of those values, with a sign set by its quadrant. Practice rebuilding it a few times, then use the quiz above to check yourself.

What is

\sin 30^\circ

on the unit circle?+

$\sin 30^\circ = 1/2$ . The 30° point on the unit circle is $(\sqrt{3}/2, 1/2)$ , and sine is the y-coordinate. In radians, 30° is $\pi /6$ , so $\sin (\pi /6) = 1/2$ as well.

Why is

\tan 90^\circ

undefined?+

Because $\tan \theta = \sin \theta / \cos \theta$ , and at 90° the point on the unit circle is (0, 1), so cos 90° = 0. Dividing by zero is undefined. The same happens at 270°, where the point is (0, −1).

What is a reference angle?+

A reference angle is the acute angle between the terminal side of an angle and the x-axis. For 150° it is 30°, for 225° it is 45°, and for 300° it is 60°. An angle has the same sine, cosine and tangent values as its reference angle, apart from the sign, which depends on the quadrant.

How do you know if sine or cosine is positive or negative?+

Look at the quadrant. Cosine is the x-coordinate, so it is positive on the right (Quadrants I and IV) and negative on the left (II and III). Sine is the y-coordinate, so it is positive above the x-axis (I and II) and negative below (III and IV). Tangent is positive where they share a sign (I and III). The memory aid "All Students Take Calculus" lists which are positive in Quadrants I to IV: all, sine, tangent, cosine.

How do you convert between degrees and radians?+

A full turn is 360° or $2\pi$ radians, so 180° = π radians. To convert degrees to radians, multiply by $\pi /180$ ; to convert radians to degrees, multiply by $180/\pi$ . For example, $135^\circ \times \pi /180 = 3\pi /4$ .

Is this unit circle tool free?+

Yes. It runs entirely in your browser, needs no signup, and stores nothing.

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