The unit circle: how to build it in two minutes instead of memorising it, and how radians become degrees

The unit circle is the circle of radius 1 centred at the origin, and the point where an angle's arm meets it has coordinates (cos θ, sin θ). You do not need to memorise 16 points. You need two triangles (the 30-60-90 and the 45-45-90, each with hypotenuse 1), the four quadrant signs, and the rule that 180° is π radians. From those you rebuild any special angle in under two minutes. Here is the build, with one angle worked line by line.

What the unit circle is, in one sentence

The unit circle is a circle of radius 1, so the coordinates of a point on it are the cosine and sine of its angle, with no division needed.

Here is why radius 1 matters. On any circle, cos θ is x divided by the radius and sin θ is y divided by the radius. Make the radius 1 and the division disappears: cos θ = x and sin θ = y. Tan θ is still a ratio, y/x, because it compares the two coordinates.

So "what is cos 150°?" becomes "how far left or right is the point at 150°?" That is a question about a picture, and a picture can be rebuilt.

In the MHF4U course this is idea 16, "Trigonometric functions of any angle and the unit circle". It comes right after idea 15, "Radian measure and arc length", and leads into idea 17, "Graphs of sine, cosine and tangent and their transformations". The MHF4U course map shows all 26 ideas in order.

Radians to degrees, and back

There is one fact to know: π radians = 180°. Every conversion follows from it.

• Radians to degrees: multiply by 180/π.
• Degrees to radians: multiply by π/180, and leave π in the answer.

Three conversions, computed:

• 5π/6 × 180/π = 5 × 180/6 = 150°
• 7π/4 × 180/π = 7 × 180/4 = 315°
• 240° × π/180 = 240π/180 = 4π/3

The factor is chosen so that π cancels. Going to degrees, π is on the top of the angle, so π must be on the bottom of the factor.

The common wrong line: multiplying by π/180 when going to degrees. Then 5π/6 becomes 5π²/1080, about 0.0457, which is not an angle anyone meant. The second mistake is replacing π with 3.14 and rounding, so the exact fraction is lost. An exact answer keeps π: 4π/3, not 4.19.

The two triangles you actually memorise

Everything on the circle comes from two right triangles, each drawn with its hypotenuse equal to 1.

The 45-45-90 triangle. Both legs are equal. With hypotenuse 1, each leg is √2/2 (about 0.707). So at 45°, the point on the circle is (√2/2, √2/2).

The 30-60-90 triangle. The short leg is 1/2 and the long leg is √3/2 (about 0.866). The short leg sits opposite the 30° angle. The long leg sits opposite the 60° angle.

So at 30°, the point is (√3/2, 1/2): a long way across, a short way up. At 60°, it is (1/2, √3/2): a short way across, a long way up.

This is the pair students swap. Use one check: the longer leg is opposite the bigger angle. A 30° arm is close to the x-axis, so the point is far across and only a little way up.

The four signs

The triangles give the sizes. The quadrant gives the signs. Since cos is x and sin is y, the signs are just the signs of the coordinates.

Quadrant	Angles	cos (x)	sin (y)	tan (y/x)
I	0° to 90°	+	+	+
II	90° to 180°	−	+	−
III	180° to 270°	−	−	+
IV	270° to 360°	+	−	−

Tan is positive where cos and sin agree, in quadrants I and III. You do not need a phrase to remember this table. Picture the point: left of the y-axis means x is negative, below the x-axis means y is negative.

Worked example: sin, cos and tan of 5π/6, line by line

Find the exact values of sin(5π/6), cos(5π/6) and tan(5π/6) without a chart.

1. Convert to degrees. 5π/6 × 180/π = 150°.
2. Find the quadrant. 150° is between 90° and 180°, so it is in quadrant II.
3. Find the reference angle. In quadrant II it is 180° − 150° = 30°, which is π/6.
4. Read the triangle. The 30° triangle gives (√3/2, 1/2) before any signs.
5. Apply the quadrant II signs. Cos is negative and sin is positive: cos(5π/6) = −√3/2 and sin(5π/6) = 1/2.
6. Divide for tan. tan(5π/6) = (1/2) ÷ (−√3/2) = −1/√3, which rationalises to −√3/3.

Three exact values, built from one triangle and one sign rule.

The common wrong line, first version: cos(5π/6) = +√3/2. The reference angle was found and the quadrant was forgotten. At 150° the point is to the left of the y-axis, so x is negative.

The common wrong line, second version: sin(5π/6) = √3/2, because 30° "sounds" like it should give the bigger value. It is the small angle, so the side opposite it is the short one, 1/2.

Sibling to try cold

Close the page above, or cover it. Find sin, cos and tan of 4π/3 without a chart, writing all six lines yourself. Trying it before you check is what makes it stick: Roediger and Karpicke (2006) found that students who tested themselves remembered more a week later than students who reread. A worked example followed by your own attempt is the order Sweller and Cooper (1985) found works fastest for a beginner.

Check

4π/3 is 240°, in quadrant III, with reference angle 60°. So sin(4π/3) = −√3/2, cos(4π/3) = −1/2 and tan(4π/3) = √3.

If you got a sign wrong, redraw the point: 240° is below and to the left of the origin. If you swapped 1/2 and √3/2, go back to the 60° triangle: the long leg is opposite 60°.

Building the whole circle in two minutes

This is how you stop needing the chart. Do it on blank paper with a timer.

1. Draw the axes and the four easy points. (1, 0) at 0, (0, 1) at π/2, (−1, 0) at π, (0, −1) at 3π/2.
2. Fill quadrant I from the two triangles. 30° at (√3/2, 1/2), 45° at (√2/2, √2/2), 60° at (1/2, √3/2).
3. Reflect into the other quadrants. Across the y-axis for quadrant II, through the origin for III, across the x-axis for IV. The sizes stay the same; change the signs by the table.
4. Label the radians by counting. The 30° family counts in sixths of π: π/6, 5π/6, 7π/6, 11π/6. The 45° family counts in quarters: π/4, 3π/4, 5π/4, 7π/4. The 60° family is π/3, 2π/3, 4π/3, 5π/3.
5. Time it, then check it. Compare with your notes and mark every wrong sign.

Be honest about what happens next: after building it ten times you will have memorised it anyway. That is the point. The memory comes from producing the circle, not from staring at it. Testing yourself before you look explains why the producing works and the staring does not.

Where it goes next

The sine and cosine graphs of idea 17 are this circle unrolled: the height of the point as the angle grows is the sine wave. The trig identities are relations between these same coordinates; sin²θ + cos²θ = 1 is just the circle's equation x² + y² = 1.

In calculus, the derivative rules evaluate sin and cos at these angles on almost every page. The precalculus ideas calculus assumes put the unit circle near the top of the list for that reason. For a course-wide method, how to study math has the routine.

If a sign error keeps costing you the mark, write it in an error log the moment you find it. That habit is chapter 4 of the free playbook. The other MHF4U explainers are listed on the math help hub.

The unit circle on one screen

• The definition: radius 1, so cos θ = x, sin θ = y, tan θ = y/x.
• The conversion: π rad = 180°. To degrees, multiply by 180/π. To radians, multiply by π/180.
• The 45° triangle: both legs √2/2.
• The 30-60-90 triangle: 1/2 opposite 30°, √3/2 opposite 60°. The longer leg is opposite the bigger angle.
• The signs: I (+, +), II (−, +), III (−, −), IV (+, −) for (cos, sin). Tan is positive in I and III.
• The method: convert, find the quadrant, find the reference angle, read the triangle, apply the signs, divide for tan.