The Unit Circle
The unit circle is the map that connects angles to coordinates, and coordinates to the trigonometric functions. Once you understand it, you can find the sine, cosine, and tangent of any angle — positive, negative, or larger than $360^\circ$ — without a calculator.
Practise this lesson
Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.
Imagine a circle with radius 1 centred at the origin. If you walk around the circumference to an angle of $45^\circ$ ($\frac{\pi}{4}$ radians), without using a calculator — what are your $x$- and $y$-coordinates? How do these coordinates relate to $\sin 45^\circ$ and $\cos 45^\circ$?
There is only one identity at the heart of this lesson. Every trig value for every angle flows from it. Lock this into memory and the rest of trigonometry becomes navigable.
If you rotate by angle $\theta$ from $(1, 0)$ on a circle of radius 1, you land at the point $(\cos \theta, \sin \theta)$. The $x$-coordinate is cosine. The $y$-coordinate is sine. The gradient is tangent.
A point at angle $\theta$ on the unit circle has coordinates $(\cos\theta, \sin\theta)$; $x = \cos\theta$ (remember: C comes before S, just as x comes before y)
Pause — copy the unit circle coordinate identity: point at angle $\theta$ has coordinates $(\cos\theta, \sin\theta)$ — $x$ = cosine (C before S, just as x before y) into your book.
Did you get this? True or false: for a point on the unit circle at angle $\theta$, the $y$-coordinate equals $\cos\theta$.
Key facts
- The definition of the unit circle
- How sine, cosine, and tangent are defined from the unit circle
- The ASTC rule for quadrant signs
- Key points on the unit circle at multiples of $\frac{\pi}{2}$
Concepts
- Why the unit circle extends trig ratios to any angle
- How reference angles simplify calculations in all quadrants
- The connection between angles and coordinates
Skills
- Find $\sin \theta$, $\cos \theta$, and $\tan \theta$ from the unit circle
- Determine the quadrant of any angle in radians
- Use reference angles to find exact values
- Apply the Pythagorean identity to find missing trig ratios
We just saw that every point on the unit circle encodes $(\cos\theta, \sin\theta)$. That raises a question: but which angles give which signs — how do we know if cosine or sine is positive or negative in each quadrant? This card answers it → the ASTC rule and the reference angle concept, which let you evaluate any angle from just the acute cases.
Unit circle: radius = 1, centred at origin — equation $x^2 + y^2 = 1$; Reference angle: always acute (between 0 and $\frac{\pi}{2}$), measured from the $x$-axis
Pause — copy the unit circle equation $x^2 + y^2 = 1$, the reference angle definition (always acute, from the $x$-axis), and the ASTC memory aid into your book.
Quick check: In which quadrant are both sine and cosine negative?
We just saw the ASTC rule for sign and the reference angle definition. That raises a question: how do reference angles combine with ASTC to let us evaluate any angle — including obtuse and reflex angles — using only the acute exact values? This card answers it → the four reference angle formulas (one per quadrant) applied to the unit circle.
The unit circle is a circle with radius $1$ centred at the origin $(0, 0)$. It is the foundation of trigonometry because it connects angles to coordinates in a simple, universal way.
If you start at the point $(1, 0)$ and rotate anticlockwise by an angle $\theta$, the point where you land is:
This means the $x$-coordinate is $\cos \theta$, the $y$-coordinate is $\sin \theta$, and the gradient of the line from the origin is $\tan \theta = \frac{\sin \theta}{\cos \theta}$.
The unit circle — radius 1, centred at the origin, showing key angles and the ASTC quadrants.
Reference angles
The reference angle $\alpha$ is the acute angle that the terminal side makes with the $x$-axis. It is always positive and between $0$ and $\frac{\pi}{2}$.
Once you know the reference angle, find the exact trig value using special triangles, then apply the correct sign from ASTC.
Unit circle point: $P(\theta) = (\cos\theta, \sin\theta)$ — x is cos, y is sin; Reference angle formulas: QI: $\alpha=\theta$, QII: $\alpha=\pi-\theta$, QIII: $\alpha=\theta-\pi$, QIV: $\alpha=2\pi-\theta$
Pause — copy the unit circle coordinate identity $P(\theta) = (\cos\theta, \sin\theta)$ and all four reference angle formulas (QI: $\alpha=\theta$; QII: $\alpha=\pi-\theta$; QIII: $\alpha=\theta-\pi$; QIV: $\alpha=2\pi-\theta$) into your book.
Fill the blanks: drag each token into the matching blank.
In QI the reference angle is ___. In QII it is ___. In QIII it is ___. In QIV it is ___.
Worked examples · 3 in a row, reveal as you go
Find the exact coordinates of the point on the unit circle corresponding to $\theta = \frac{2\pi}{3}$.
If $\sin \theta = -\frac{3}{5}$ and $\theta$ is in Quadrant IV, find $\cos \theta$.
Find the exact value of $\cos\left(-\frac{\pi}{4}\right)$.
Follow-up: Using the same approach, what is the exact value of $\sin\!\left(-\frac{\pi}{6}\right)$?
Common errors · the 3 traps that cost marks
Odd one out: Three of these statements are correct. Which one is WRONG?
Quick-fire practice · 5 reps
Find the coordinates on the unit circle for $\theta = \pi$.
Find the coordinates on the unit circle for $\theta = \frac{5\pi}{4}$.
Find the coordinates on the unit circle for $\theta = -\frac{\pi}{2}$.
Find the coordinates on the unit circle for $\theta = \frac{5\pi}{6}$.
State the exact value of $\tan \frac{4\pi}{3}$.
Earlier you were asked: What are your coordinates at $45^\circ$ on the unit circle, and how do they relate to $\sin 45^\circ$ and $\cos 45^\circ$?
At $\theta = \frac{\pi}{4}$ ($45^\circ$), the point on the unit circle is $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$. By the unit circle definition, the $x$-coordinate is $\cos \theta$ and the $y$-coordinate is $\sin \theta$. Therefore $\cos 45^\circ = \frac{\sqrt{2}}{2}$ and $\sin 45^\circ = \frac{\sqrt{2}}{2}$. This is the only acute angle where sine and cosine are exactly equal.
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Q1. (a) State the exact coordinates of the point on the unit circle corresponding to $\theta = \frac{3\pi}{4}$. (b) Hence, write down the exact values of $\sin \frac{3\pi}{4}$ and $\cos \frac{3\pi}{4}$. 3 MARKS
Q2. If $\cos \theta = -\frac{5}{13}$ and $\theta$ is in Quadrant II, find the exact value of $\sin \theta$. Show your working. 2 MARKS
Q3. A student claims that $\sin \theta$ is always positive when $\theta$ is between $0$ and $\pi$, and always negative when $\theta$ is between $\pi$ and $2\pi$. Evaluate this claim, using specific examples from the unit circle to support your answer. 3 MARKS
Comprehensive answers (click to reveal)
Drill 1: $(-1, 0)$
Drill 2: $\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$
Drill 3: $(0, -1)$
Drill 4: $\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$
Drill 5: QIII, reference $\frac{\pi}{3}$, tan positive. $\tan \frac{4\pi}{3} = \sqrt{3}$
Q1 (3 marks): (a) $\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$ [1]. (b) $\sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2}$ [1], $\cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2}$ [1].
Q2 (2 marks): $\sin^2 \theta = 1 - \frac{25}{169} = \frac{144}{169}$ [1]. In QII, sine is positive, so $\sin \theta = \frac{12}{13}$ [1].
Q3 (3 marks): The claim is partially correct but imprecise [0.5]. Between $0$ and $\pi$, sine is positive in QI and QII, so the first part is correct [0.5]. Between $\pi$ and $2\pi$, sine is negative in QIII and QIV, so the second part is also correct [0.5]. But the claim ignores that $\sin \pi = 0$ and $\sin 2\pi = 0$, which are neither positive nor negative [1]. The claim is broadly true for open intervals $(0, \pi)$ and $(\pi, 2\pi)$, but false at the endpoints [0.5].
Five timed questions. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
Enter the arenaClimb platforms by answering unit circle questions. Lighter alternative to the boss.
Mark lesson as complete
Tick when you've finished the practice and review.