Angles all the way round
The unit circle is simply a circle of radius 1, centred on the origin of a set of axes. You measure angles from the positive \(x\)-axis, turning anticlockwise for a positive angle. As a point travels round the edge, its coordinates are exactly \((\cos\theta,\ \sin\theta)\), and that is the whole idea: cos is the across (horizontal) distance and sin is the up (vertical) distance.
- Quadrant
- One of the four quarters the axes cut the circle into. The first quadrant runs from \(0^\circ\) to \(90^\circ\), the second from \(90^\circ\) to \(180^\circ\), the third from \(180^\circ\) to \(270^\circ\) and the fourth from \(270^\circ\) to \(360^\circ\).
Because sin and cos are read straight off the coordinates, this picture works for angles well beyond the \(0^\circ\) to \(90^\circ\) you met in right-angled triangles. An angle of \(210^\circ\) lands in the third quadrant, where both coordinates are negative, so both \(\sin 210^\circ\) and \(\cos 210^\circ\) come out negative.
Degrees and radians
Degrees are one way to measure a turn, but the other unit, the radian, is the one that makes later work tidier. A full turn is \(360^\circ\), and that same full turn is \(2\pi\) radians. That single fact converts everything:
So the common angles convert to \(30^\circ=\tfrac{\pi}{6}\), \(45^\circ=\tfrac{\pi}{4}\), \(60^\circ=\tfrac{\pi}{3}\) and \(90^\circ=\tfrac{\pi}{2}\). To go the other way, multiply the radian value by \(\tfrac{180}{\pi}\).
Keep the pi
When converting to radians, leave the answer as an exact multiple of \(\pi\), such as \(\tfrac{2\pi}{3}\), rather than typing it as a decimal. Exact form is what most mark schemes want.
The CAST sign rule
The quadrant a point sits in decides whether each ratio is positive or negative. The word CAST records this, read anticlockwise starting from the fourth quadrant at the bottom right:
| Quadrant | Angles | Positive ratios |
|---|---|---|
| First (A) | \(0^\circ\) to \(90^\circ\) | All: sin, cos, tan |
| Second (S) | \(90^\circ\) to \(180^\circ\) | Sin only |
| Third (T) | \(180^\circ\) to \(270^\circ\) | Tan only |
| Fourth (C) | \(270^\circ\) to \(360^\circ\) | Cos only |
To find an exact value for any angle, pair CAST with the reference angle: the acute angle to the nearest part of the \(x\)-axis. The reference angle gives the size, and CAST gives the sign.
Find the exact value of \(\cos 150^\circ\).
Exact values to memorise
These five angles turn up constantly, and you are expected to know their ratios without a calculator. It is worth learning this table until it is automatic:
| \(\theta\) | \(\sin\theta\) | \(\cos\theta\) | \(\tan\theta\) |
|---|---|---|---|
| \(0^\circ\) | \(0\) | \(1\) | \(0\) |
| \(30^\circ\) | \(\tfrac{1}{2}\) | \(\tfrac{\sqrt{3}}{2}\) | \(\tfrac{1}{\sqrt{3}}\) |
| \(45^\circ\) | \(\tfrac{\sqrt{2}}{2}\) | \(\tfrac{\sqrt{2}}{2}\) | \(1\) |
| \(60^\circ\) | \(\tfrac{\sqrt{3}}{2}\) | \(\tfrac{1}{2}\) | \(\sqrt{3}\) |
| \(90^\circ\) | \(1\) | \(0\) | undefined |
A couple of patterns make this easier to hold on to. The sin column climbs \(0, \tfrac12, \tfrac{\sqrt2}{2}, \tfrac{\sqrt3}{2}, 1\), and the cos column is just that same list read backwards. Each tangent is simply \(\tan\theta = \dfrac{\sin\theta}{\cos\theta}\), which is why \(\tan 90^\circ\) is undefined: you would be dividing by \(\cos 90^\circ = 0\).
Where this is assessed
The unit circle is pure Criterion A (knowing and applying the sign rule and exact values) and Criterion C (writing answers in exact form with correct notation). A rounded decimal where an exact surd was asked for will drop marks.
Check yourself
1. In which quadrant does \(210^\circ\) lie, and is \(\sin 210^\circ\) positive or negative? +
\(210^\circ\) is between \(180^\circ\) and \(270^\circ\), so it is in the third quadrant. By CAST only tan is positive there, so \(\sin 210^\circ\) is negative.
2. Find the exact value of \(\tan 225^\circ\). +
\(225^\circ\) is in the third quadrant, where tan is positive. The reference angle is \(225^\circ - 180^\circ = 45^\circ\), and \(\tan 45^\circ = 1\). So \(\tan 225^\circ = \mathbf{1}\).
3. Convert \(120^\circ\) to radians, leaving your answer in terms of \(\pi\). +
Multiply by \(\tfrac{\pi}{180}\): \(120 \times \tfrac{\pi}{180} = \tfrac{120\pi}{180} = \mathbf{\tfrac{2\pi}{3}}\).
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