Radian measure, arc length and sector area, the trigonometric ratios and their graphs, exact values, identities, the reciprocal and inverse functions, the addition and double angle formulae, and solving trigonometric equations.

What this dot point is asking

AQA wants you to work in radians, use the arc length and sector area formulae, know the graphs and exact values of the trigonometric functions, apply the Pythagorean, addition and double angle identities, use the reciprocal and inverse functions, and solve trigonometric equations over given intervals. Equation solving, often after rewriting with an identity or in harmonic form, is the most heavily examined skill here.

Radians, arc length and sector area

A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. Since the full circumference is $2\pi r$, a full turn is $2\pi$ radians, so $\pi$ radians equals $180$ degrees. Convert degrees to radians by multiplying by $\frac{\pi}{180}$.

Exact values and graphs

You must know the exact values such as $\sin 30^\circ = \frac{1}{2}$, $\cos 30^\circ = \frac{\sqrt{3}}{2}$, $\sin 45^\circ = \frac{1}{\sqrt 2}$ and $\tan 45^\circ = 1$, and their radian equivalents. The graphs of $\sin x$ and $\cos x$ have period $2\pi$ and range $-1$ to $1$; $\tan x$ has period $\pi$ with vertical asymptotes at odd multiples of $\frac{\pi}{2}$. Knowing the shape and symmetry of these graphs is what lets you find all solutions to an equation.

Identities

The reciprocal functions are $\sec\theta = \frac{1}{\cos\theta}$, $\csc\theta = \frac{1}{\sin\theta}$ and $\cot\theta = \frac{1}{\tan\theta}$. The inverse functions $\arcsin$, $\arccos$ and $\arctan$ are defined on restricted domains so that they are one-to-one and single-valued.

Solving trigonometric equations

The reliable method is: reduce to a single trigonometric function (using an identity if both sine and cosine appear), solve for that function, find the principal value, then use the graph symmetry to list every solution in the interval.

The harmonic form $a\sin\theta + b\cos\theta = R\sin(\theta + \alpha)$, where $R = \sqrt{a^2 + b^2}$ and $\tan\alpha = \frac{b}{a}$, rewrites an expression with both sine and cosine as a single sine, which makes equations and maximum or minimum values straightforward. Because the single sine ranges between $-R$ and $R$, the maximum value of the expression is $R$ (when the sine equals one) and the minimum is $-R$, and you can read off the value of $\theta$ at which each occurs. This is a common follow-up part after expressing in harmonic form.

When the equation involves a multiple angle, such as $\sin 2x = 0.5$ over $0 \le x \le 2\pi$, first widen the interval for the inner angle (here $0 \le 2x \le 4\pi$), solve for $2x$ across that wider range to capture every solution, then divide by the multiple at the end. Forgetting to widen the interval is the single most common cause of lost solutions in multiple-angle equations.