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The unit circle explains where every exact trigonometric value comes from. Once you understand it, you no longer need to memorise a table of values.
This guide explains what the unit circle is, how to read it, how to find exact values and the mistakes to avoid.
The unit circle is a circle with a radius of 1, centred at the origin.
Take an angle θ, measured anticlockwise from the positive x-axis. Draw a radius at that angle. The point where the radius meets the circle has the coordinates (cos θ, sin θ).
This comes straight from the right-angled triangle definitions. In a right-angled triangle, cos θ is the adjacent side divided by the hypotenuse, and sin θ is the opposite side divided by the hypotenuse. On the unit circle, the hypotenuse is the radius, which is 1. So cos θ equals the x-coordinate and sin θ equals the y-coordinate.
Two results follow from this.
Measure angles anticlockwise from the positive x-axis. The x-coordinate gives you cos θ. The y-coordinate gives you sin θ.
| Angle | Radians | Coordinates (cos θ, sin θ) | What to notice |
|---|---|---|---|
| 0° | 0 | (1, 0) | cos θ is at its maximum and sin θ is 0 |
| 30° | π/6 | (√3/2, 1/2) | sin θ is small because the angle is small |
| 45° | π/4 | (√2/2, √2/2) | Both values are equal |
| 60° | π/3 | (1/2, √3/2) | The 30° values swap places |
| 90° | π/2 | (0, 1) | sin θ is at its maximum and cos θ is 0 |
| 180° | π | (−1, 0) | cos θ is at its minimum |
| 270° | 3π/2 | (0, −1) | sin θ is at its minimum |
| 360° | 2π | (1, 0) | You are back at the start |
The values for 30° and 60° are the same pair in reverse order. This is because 30° and 60° add to 90°, so they come from the same triangle. If you remember the 30° values, you also know the 60° values.
The radius of the unit circle is 1. No point on the circle can be more than 1 unit from the origin in any direction. So neither coordinate can be greater than 1.
This explains why your calculator shows an error for sin⁻¹(2). No angle has a sine of 2, because no point on the unit circle has a y-coordinate of 2.
Tangent is different. tan θ = sin θ ÷ cos θ, so it divides one coordinate by the other. As cos θ gets close to 0, tan θ gets very large. At 90° and 270°, cos θ = 0, so tan θ is undefined. This is why the graph of y = tan x has asymptotes at these angles.
The axes split the unit circle into four quadrants. The signs of the x and y coordinates in each quadrant tell you the signs of cos θ, sin θ and tan θ.
| Quadrant | Angles | cos θ (x) | sin θ (y) | tan θ | Positive ratios |
|---|---|---|---|---|---|
| First | 0° to 90° | + | + | + | All |
| Second | 90° to 180° | − | + | − | Sine |
| Third | 180° to 270° | − | − | + | Tangent |
| Fourth | 270° to 360° | + | − | − | Cosine |
Many students remember the last column with the phrase “All Stations To Central”. Read it anticlockwise from the first quadrant. All, Sine, Tangent, Cosine.
You can also work out each sign directly. In the second quadrant, points lie to the left of the y-axis, so x is negative and cos θ is negative. The points lie above the x-axis, so y is positive and sin θ is positive. tan θ divides a positive by a negative, so it is negative.
In the third quadrant, both coordinates are negative. A negative divided by a negative is positive, so tan θ is positive. Students often get this quadrant wrong when they guess.
You can find every exact value you need from two triangles. Both use surds and Pythagoras’ theorem.
Draw a square with sides of 1. Cut it along the diagonal. You get a right-angled triangle with two sides of 1 and a hypotenuse of √2.
So sin 45° = cos 45° = 1/√2. Rationalise the denominator to get √2/2.
Draw an equilateral triangle with sides of 2. Draw a line from the top corner straight down to the base. You get a right-angled triangle with a hypotenuse of 2, a base of 1 and a height of √3.
So sin 30° = 1/2 and cos 30° = √3/2. The values for 60° are the same pair swapped, so sin 60° = √3/2 and cos 60° = 1/2.
Follow these steps for any angle outside the first quadrant.
So cos 210° = −√3/2.
So sin 135° = √2/2.
A right-angled triangle can only contain angles between 0° and 90°. The unit circle has no such limit. You can keep rotating past 360°, or rotate clockwise to make negative angles. Every rotation still lands on a point with coordinates.
This turns sin θ and cos θ into functions you can use for any angle. It also explains why they repeat every 360°, or 2π radians.
When you plot these values against the angle, you get the sine and cosine graphs. In Year 12, you use these functions in calculus. For example, the derivative of sin x is cos x.
Students who understand the unit circle usually find the identities, the graphs and the calculus easier to learn.
| The mistake | What goes wrong | How to fix it |
|---|---|---|
| Mixing up the coordinates | Reading sin θ from the x-axis | x is cos θ and y is sin θ. Both c and x come before s and y in the alphabet. |
| Measuring clockwise | The signs come out wrong | Measure positive angles anticlockwise from the positive x-axis. |
| Guessing the sign in the third quadrant | Assuming tan θ is negative | Both coordinates are negative, so tan θ is positive. |
| Memorising the table without the triangles | Swapping the 30° and 60° values | Draw the half-equilateral triangle and read the values from it. |
| Using the wrong calculator mode | Answers in degrees when the question uses radians | Check the calculator mode before each question. |
| Skipping the diagram | Sign errors | Sketch the unit circle for every question. |
| Forgetting tan θ is undefined at 90° | Treating a calculator error as your mistake | cos 90° = 0, so tan 90° has no value. |
In NSW, students start learning the unit circle in Year 11 Mathematics Advanced, as part of trigonometric functions. They use it throughout Year 12 and in Mathematics Extension 1 and Extension 2. You can read the course content in the Mathematics Advanced 11-12 Syllabus on the NSW Curriculum website.
The unit circle uses surds, coordinates and Pythagoras’ theorem together. Students who are unsure of these skills from Year 9 and Year 10 Maths often find this topic hard.
Our Year 11 Maths Advanced course covers Trigonometry in Matrix Term 2 (January to March 2027) and Further Trigonometry in Matrix Term 3 (April to June 2027). Each term has nine weekly 3-hour lessons with experienced teachers, in small classes on campus in Sydney or online. Term courses cost $1,494 per term.
The unit circle is a circle with a radius of 1, centred at the origin. For any angle θ measured anticlockwise from the positive x-axis, the point on the circle has the coordinates (cos θ, sin θ).
With a radius of 1, the hypotenuse of each triangle is 1. Dividing by 1 changes nothing, so the x-coordinate equals cos θ and the y-coordinate equals sin θ.
It is a memory aid for which ratios are positive in each quadrant. Read it anticlockwise from the first quadrant. All three ratios are positive in the first quadrant, sine in the second, tangent in the third and cosine in the fourth.
Draw a square with sides of 1 and cut it along the diagonal for the 45° values. Draw an equilateral triangle with sides of 2 and split it in half for the 30° and 60° values. For other angles, use the reference angle and the sign for the quadrant.
A free trial lesson lets you learn trigonometry with an experienced Matrix Maths teacher. Trial lessons run at our seven Sydney campuses. Free trials do not include Holiday Courses or Matrix+.
Written by Ruby
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