Revise The unit circle and radian measure for IB Mathematics: Analysis and Approaches (3.3)

A radian is defined by arc length. One radian is the angle at the centre of a circle for which the arc on the circumference has the same length as the radius. Since the full circumference is $2\pi r$, a complete turn wraps $2\pi$ radii around the circle, so a full circle is $2\pi$ radians. Setting that equal to $360^{\circ}$ gives the one relationship everything else is built on.

$2\pi \text{ rad} = 360^{\circ} \quad\Longrightarrow\quad \pi \text{ rad} = 180^{\circ}$

A sector of angle $\theta$ is the fraction $\tfrac{\theta}{2\pi}$ of a full circle. Multiplying that fraction by the whole circumference $2\pi r$ gives the arc length, and by the whole area $\pi r^2$ gives the sector area. The $2\pi$ cancels beautifully — but only because $\theta$ is in radians.

Arc length: $\; s = r\theta \qquad\qquad$ Sector area: $\; A = \tfrac{1}{2}r^2\theta \qquad (\theta \text{ in radians})$

The unit circle is the circle of radius 1 centred at the origin. From Pythagoras its equation is $x^2 + y^2 = 1$. Take any point $P$ on it and let $\theta$ be the angle measured anticlockwise from the positive x-axis to the radius $OP$. Then the coordinates of $P$ are defined to be the cosine and sine of that angle.

For $P(x, y)$ on the unit circle at angle $\theta$: $\quad x = \cos\theta, \quad y = \sin\theta, \quad \tan\theta = \dfrac{\sin\theta}{\cos\theta} = \dfrac{y}{x}$

This is a genuine extension of the right-angled-triangle definitions. For an acute angle it agrees with opposite-over-hypotenuse and so on, because the hypotenuse here is the radius 1. But the coordinate definition keeps working for angles beyond $90^{\circ}$, for negative angles, and for full turns and more — the point simply keeps moving round the circle. The tangent is the gradient of $OP$, which is why it blows up to undefined whenever $OP$ is vertical and $\cos\theta = 0$.

On Paper 1 you cannot reach for a calculator, so the exact values for the special first-quadrant angles must be instant recall. They come from two triangles — the half-square (for $\tfrac{\pi}{4}$) and the half-equilateral triangle (for $\tfrac{\pi}{6}$ and $\tfrac{\pi}{3}$) — together with the axis cases $0$ and $\tfrac{\pi}{2}$.

A helpful memory aid: for $\sin$ at $0, \tfrac{\pi}{6}, \tfrac{\pi}{4}, \tfrac{\pi}{3}, \tfrac{\pi}{2}$ write $\tfrac{\sqrt{0}}{2}, \tfrac{\sqrt{1}}{2}, \tfrac{\sqrt{2}}{2}, \tfrac{\sqrt{3}}{2}, \tfrac{\sqrt{4}}{2}$ — the numerators are just $\sqrt{0}$ up to $\sqrt{4}$. Cosine is the same list read backwards. Once you have $\sin$ and $\cos$, every $\tan$ is their quotient.

Because $\cos\theta$ is the x-coordinate and $\sin\theta$ the y-coordinate, the sign of each ratio is simply the sign of that coordinate in the quadrant. Collecting the results gives the CAST rule, read anticlockwise from the first quadrant.

Symmetry turns one known value into four. If $\alpha$ is the reference (acute) angle, then across the circle the magnitude of each ratio is the value at $\alpha$, and CAST supplies the sign. For instance, knowing $\sin\tfrac{\pi}{3} = \tfrac{\sqrt{3}}{2}$ immediately gives $\sin\tfrac{2\pi}{3} = +\tfrac{\sqrt{3}}{2}$ (Q2, sine positive) and $\sin\tfrac{4\pi}{3} = -\tfrac{\sqrt{3}}{2}$ (Q3, sine negative). You never learn more than the first quadrant.

In IB Mathematics the marks are analytic. A method mark (M) is awarded for a correct approach — the right formula, correctly used — and is not lost to an arithmetic slip later. An accuracy mark (A) is for a correct answer, but many A-marks are dependent on the corresponding M: no valid method, no accuracy mark. Two more conventions matter here. Follow-through (FT) means that if you carry a wrong earlier value into a correct later step, the later mark can still be earned on your own figure. And 'accept equivalent or exact forms' means the engine takes any correct algebraic form — $5\pi$ cm or $15.7$ cm, $\tfrac{1}{\sqrt{2}}$ or $\tfrac{\sqrt{2}}{2}$ — provided it is right. Study how each mark below is tied to one specific line of working.

The unit circle is the launch pad for the whole trigonometry strand. Reading $\sin\theta$ and $\cos\theta$ as coordinates as the point travels round the circle is exactly what generates the trigonometric graphs and their periods in 3.5, and the symmetry you used for related angles becomes the identities and equations of 3.6 and beyond. Radians, meanwhile, are non-negotiable for the calculus of trigonometric functions later in the course, where $\tfrac{d}{dx}\sin x = \cos x$ holds only because $x$ is in radians. Master the habits here — check the angle's units, keep values exact, apply CAST every time, and show each line of method — and the trigonometry that follows becomes variations on a picture you already own.