The unit circle is the quickest way to see what sin⁡θ\sin \theta, cos⁡θ\cos \theta, and tan⁡θ\tan \theta mean. It is the circle centered at the origin with radius 11, and the point reached by an angle θ\theta is

(cos⁡θ,sin⁡θ)(\cos \theta, \sin \theta)

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

tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}

On a unit circle, radian measure has a special meaning: because the radius is 11, the arc length from 00 to θ\theta is exactly θ\theta radians.

Explore The Unit Circle

Move the angle and watch the point, the coordinates, and the trig values change together. Start with 30∘30^\circ, 150∘150^\circ, 210∘210^\circ, and 330∘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.

xy(0.707, 0.707)cos(theta)sin(theta)45 deg
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⁡θ,sin⁡θ)(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 xx-axis have negative sine.

  • Quadrant I: sin⁡θ>0\sin \theta > 0 and cos⁡θ>0\cos \theta > 0
  • Quadrant II: sin⁡θ>0\sin \theta > 0 and cos⁡θ<0\cos \theta < 0
  • Quadrant III: sin⁡θ<0\sin \theta < 0 and cos⁡θ<0\cos \theta < 0
  • Quadrant IV: sin⁡θ<0\sin \theta < 0 and cos⁡θ>0\cos \theta > 0

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

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

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

At 30∘30^\circ, the point is

(32,12)\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)

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

(cos⁡150∘,sin⁡150∘)=(−32,12)(\cos 150^\circ, \sin 150^\circ) = \left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)

Then

tan⁡150∘=sin⁡150∘cos⁡150∘=1/2−3/2=−13\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∘30^\circ and 150∘150^\circ. The coordinate magnitudes stay the same, but the quadrant flips the sign of the xx-coordinate.

What To Notice In The Explorer

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

  1. Angles with the same reference angle reuse the same coordinate magnitudes.
  2. Adding 360∘360^\circ lands on the same point, so sine and cosine repeat every full turn.
  3. The axes are boundary cases where one coordinate becomes 00.

Try A Similar Check

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

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