Unit Circle

The unit circle is the quickest way to see what $\sin \theta$ , $\cos \theta$ , and $\tan \theta$ mean. It is the circle centered at the origin with radius $1$ , and the point reached by an angle $\theta$ is

(\cos \theta, \sin \theta)

So cosine is the horizontal coordinate and sine is the vertical coordinate. Tangent comes from their ratio when $\cos \theta \ne 0$ :

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

On a unit circle, radian measure has a special meaning: because the radius is $1$ , the arc length from $0$ to $\theta$ is exactly $\theta$ radians.

Explore The Unit Circle

Move the angle and watch the point, the coordinates, and the trig values change together. Start with $30^\circ$ , $150^\circ$ , $210^\circ$ , and $330^\circ$ to see how one reference angle can produce four different sign patterns.

Unit circle explorer

Move the angle and compare three linked ideas: the point on the circle, the coordinates, and the trig values. The x-coordinate is cos(theta), the y-coordinate is sin(theta), and coterminal angles land on the same point.

Angle in degreesCurrent angle: 45 deg (0.125 turns)Type a degree valueNegative angles rotate clockwise. Positive angles rotate counterclockwise.

What to notice

The point always stays one unit from the origin, so its coordinates satisfy x^2 + y^2 = 1. Moving around the circle changes cos(theta) and sin(theta), but not the radius.

If you add or subtract 360 deg, the point does not move. In this view, 45 deg is already in standard position.

Current values

Actual angle: 45 deg

Radian measure: pi/4

Standard position: 45 deg

Quadrant or axis: Quadrant I

Reference angle: 45 deg

Point on the circle: (0.7071, 0.7071)

cos(theta): 0.7071

sin(theta): 0.7071

tan(theta): 1

x^2 + y^2: 1

Special-angle check

Normalized special angle: 45 deg

Equivalent radian position in one turn: pi/4

Exact point: (sqrt(2)/2, sqrt(2)/2)

Exact cos(theta): sqrt(2)/2

Exact sin(theta): sqrt(2)/2

Exact tan(theta): 1

Try this

Compare 30 deg, 150 deg, 210 deg, and 330 deg. The reference angle stays 30 deg, so the absolute values of the coordinates match while the signs change by quadrant.

Read Sine And Cosine From The Coordinates

Every point on the circle has the form $(x, y) = (\cos \theta, \sin \theta)$ . That means you do not need a separate rule for sine and cosine: just read the horizontal and vertical coordinates of the point.

This also gives a fast sign check. Points on the left half have negative cosine, and points below the $x$ -axis have negative sine.

Quadrant I: $\sin \theta > 0$ and $\cos \theta > 0$
Quadrant II: $\sin \theta > 0$ and $\cos \theta < 0$
Quadrant III: $\sin \theta < 0$ and $\cos \theta < 0$
Quadrant IV: $\sin \theta < 0$ and $\cos \theta > 0$

Tangent is undefined when $\cos \theta = 0$ . On the unit circle, that happens at the top and bottom points: $90^\circ$ and $270^\circ$ .

Worked Example: $150^\circ$ On The Unit Circle

$150^\circ$ lies in Quadrant II, so cosine should be negative and sine should be positive. Its reference angle is $30^\circ$ , which means the coordinate magnitudes match the $30^\circ$ point.

At $30^\circ$ , the point is

\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)

Quadrant II changes only the sign of the $x$ -coordinate, so at $150^\circ$ :

(\cos 150^\circ, \sin 150^\circ) = \left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)

Then

\tan 150^\circ = \frac{\sin 150^\circ}{\cos 150^\circ} = \frac{1/2}{-\sqrt{3}/2} = -\frac{1}{\sqrt{3}}

Use the explorer to jump between $30^\circ$ and $150^\circ$ . The coordinate magnitudes stay the same, but the quadrant flips the sign of the $x$ -coordinate.

What To Notice In The Explorer

Use the widget to test a few patterns instead of memorizing isolated facts:

Angles with the same reference angle reuse the same coordinate magnitudes.
Adding $360^\circ$ lands on the same point, so sine and cosine repeat every full turn.
The axes are boundary cases where one coordinate becomes $0$ .

Try A Similar Check

Pick one angle in each quadrant and predict the signs of $\sin \theta$ and $\cos \theta$ before checking the widget. Then choose a reference angle such as $30^\circ$ or $45^\circ$ and see how the same coordinate magnitudes reappear around the circle.

Frequently Asked Questions

What is the unit circle?: The unit circle is the circle centered at the origin with radius 1. The point reached by an angle theta has coordinates equal to cosine theta and sine theta, so cosine is the horizontal coordinate and sine is the vertical coordinate. Because the radius is 1, the arc length from 0 to theta is exactly theta radians.
How do you read sine and cosine from the unit circle?: Every point on the circle has the form x equals cosine theta and y equals sine theta, so just read the horizontal coordinate for cosine and the vertical coordinate for sine. This also gives a fast sign check: points on the left half have negative cosine, and points below the x axis have negative sine.
Where is tangent undefined on the unit circle?: Tangent equals sine divided by cosine, so it is undefined wherever cosine is zero. On the unit circle, that happens at the top and bottom points, which correspond to 90 degrees and 270 degrees. Everywhere else, you can compute tangent directly from the coordinates of the point.
What are the signs of sine and cosine in each quadrant?: In Quadrant I both sine and cosine are positive. In Quadrant II sine is positive and cosine is negative. In Quadrant III both are negative. In Quadrant IV sine is negative and cosine is positive. Knowing the quadrant tells you the signs before you compute any values.
How do you find the trig values of 150 degrees using a reference angle?: 150 degrees lies in Quadrant II, where cosine is negative and sine is positive, and its reference angle is 30 degrees. The coordinate magnitudes match the 30 degree point, so cosine of 150 degrees is negative square root of 3 over 2 and sine is one half. Tangent is their ratio, negative 1 over square root of 3.

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