Trigonometry

Trigonometry is the heaviest Paper 01 topic (9 of 45 items) and a recurring element of Paper 02. The section builds from exact values and graphs through to identity proofs and equation solving, all using radians as the default angle unit.

Radians¶

One radian is the angle subtended at the centre of a circle when the arc length equals the radius.

$180° = \pi \text{ radians}$

Degrees to radians: multiply by $\dfrac{\pi}{180}$

Radians to degrees: multiply by $\dfrac{180}{\pi}$

Arc Length and Sector Area¶

For a circle of radius $r$ and sector angle $\theta$ in radians:

$\text{Arc length:} \quad s = r\theta \qquad \text{Sector area:} \quad A = \frac{1}{2}r^2\theta$

Exact Trigonometric Values¶

These must be memorised. They derive from the equilateral triangle (for $30°, 60°$) and the isosceles right triangle (for $45°$).

For related angles in other quadrants, use the CAST diagram.

The CAST Diagram¶

The sign of each trig function depends on the quadrant. CAST tells you which functions are positive in each quadrant (all others are negative):

Related angle method: For any angle $\theta$ outside $[0°, 90°]$, find the reference angle (acute angle to the nearest $x$-axis), evaluate the trig function for that acute angle, then apply the sign from CAST.

Graphs of Trigonometric Functions¶

For $y = \sin kx$, $y = \cos kx$, and $y = \tan kx$:

Increasing $k$ compresses the graph horizontally (shorter period). The $\sin$ and $\cos$ graphs are identical in shape; $\cos$ is $\sin$ shifted $\frac{\pi}{2}$ to the left.

For $k = 2$ specifically: $y = \sin 2x$ and $y = \cos 2x$ each complete two full cycles in $[0, 2\pi]$, so their period is $\pi$. This is the most commonly tested value of $k$ beyond $k = 1$.

Trigonometric Identities¶

The fundamental Pythagorean identity derives from the unit circle:

$\sin^2\theta + \cos^2\theta \equiv 1$

Dividing through by $\cos^2\theta$:

$\tan^2\theta + 1 \equiv \sec^2\theta$

The identity $\tan\theta \equiv \dfrac{\sin\theta}{\cos\theta}$ is equally essential.

Compound-Angle Formulas¶

$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$

$\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$

$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$

Note the sign pattern: in $\cos(A+B)$ the right-hand side uses minus, and in $\cos(A-B)$ it uses plus (opposite to the left-hand side). Students are not expected to prove these formulas, but must apply them accurately.

Double-Angle Formulas¶

Setting $B = A$ in the compound-angle formulas gives:

$\sin 2A = 2\sin A\cos A$

$\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A$

$\tan 2A = \frac{2\tan A}{1 - \tan^2 A}$

The three forms of $\cos 2A$ are all equivalent (each comes from substituting $\sin^2 A + \cos^2 A = 1$ into the first). Which to use depends on what the question asks for: if you need $\cos 2A$ in terms of $\sin A$ only, use $1 - 2\sin^2 A$; in terms of $\cos A$ only, use $2\cos^2 A - 1$; if both appear, the first form $\cos^2 A - \sin^2 A$ is most flexible.

Proving Trigonometric Identities¶

An identity is true for all valid values of the variable. To prove it, work on one side only (usually the more complex side) and transform it step by step until it matches the other side. The reason you cannot move terms across the equals sign is that doing so would assume the identity is already true before you have proved it, which is circular reasoning. The goal is to show both sides are equal independently.

Solving Trigonometric Equations¶

The syllabus requires solutions in $0 \leq \theta \leq 2\pi$ (radians). General solutions are not required.

Standard method:

1. Isolate the trig function (e.g. $\sin\theta = \frac{1}{2}$).
2. Find the reference angle using exact values or a calculator.
3. Use CAST to identify which quadrants give solutions in the required range.
4. List all solutions.

Equations Using Identities¶

Many trig equations require substituting an identity to reduce to a standard form.