Use trigonometry: apply the sine rule, cosine rule and area formula, use the identities for tan and the Pythagorean identity, and solve trigonometric equations over a given interval.

What this dot point is asking

Trigonometry in CCEA GCSE Further Mathematics extends the right-angled work of ordinary GCSE to any triangle and to equations. You must use the sine rule, the cosine rule and the area formula for non-right-angled triangles, know and use the tan identity and the Pythagorean identity to simplify and prove, and solve trigonometric equations across a stated interval, finding every solution. These skills support the coordinate-geometry and calculus topics and appear on both calculator and exact-value questions.

The sine and cosine rules

The sine and cosine rules handle triangles that are not right-angled. Use the sine rule when you have a matching side and angle pair; use the cosine rule when you have two sides and the included angle, or all three sides.

Choosing the right rule is half the marks. If the angle sits between the two sides you know, the cosine rule finds the third side; if you can pair a side with its opposite angle, the sine rule is quicker.

Exact values and the unit-circle behaviour

You should know the exact values of sine, cosine and tangent for the special angles $0^\circ, 30^\circ, 45^\circ, 60^\circ$ and $90^\circ$, such as $\sin 30^\circ = \tfrac{1}{2}$, $\cos 30^\circ = \tfrac{\sqrt{3}}{2}$ and $\tan 45^\circ = 1$. You also need the signs of each ratio in each quadrant, which determine where extra solutions to an equation lie: sine is positive for $0^\circ$ to $180^\circ$, cosine is positive for $0^\circ$ to $90^\circ$ and $270^\circ$ to $360^\circ$, and tangent repeats every $180^\circ$.

Trigonometric identities

The two identities let you rewrite expressions and prove results. The tan identity turns a ratio of sine and cosine into a tangent, and the Pythagorean identity links the squares of sine and cosine.

These are used both to simplify (replacing $1 - \cos^2\theta$ by $\sin^2\theta$) and to convert an equation into one in a single ratio so it can be solved.

Solving trigonometric equations

A trigonometric equation usually has several solutions in an interval, because the graphs repeat. Find the first value, then use the graph's symmetry to find the rest.

Why this matters

Trigonometry is essential across the qualification. The sine and cosine rules solve real geometry and feed the Mechanics unit through resolving forces and velocities; the identities streamline proofs and equation solving; and exact values keep non-calculator answers tidy. Solving equations correctly over an interval, finding every root, is a frequently tested and frequently lost set of marks.