Trigonometric methods (internal)
Internal 4 credits Radians Trig equations Identities Sinusoidal models Sine & cosine rulesAS 91575, 4 credits. Apply trigonometric methods in solving problems — equations, identities and, almost always, a sinusoidal model fitted to a real cycle (tides, daylight, temperature, a Ferris wheel). The modelling part is where relational and extended abstract thinking are easiest to earn.
1 Reading the notation
Radians, exact values and the CAST rule — get these fast and automatic before anything else.📖 Terms and tools used throughout
4 terms- Radian
- The angle where arc length equals the radius; π radians = 180°.
- CAST rule
- Which trig function is positive in each quadrant, going anticlockwise from quadrant 1: All, Sin, Tan, Cos.
- Period
- The horizontal distance after which the function repeats — 2π for sin/cos, π for tan.
- Amplitude
- Half the vertical distance between the maximum and minimum of a sinusoidal graph.
2 What the grades require
Same ladder for all three Calculus internals.| Grade | What it takes | The half people drop |
|---|---|---|
| Achieved | Select and use methods; know the concepts and terms; communicate with appropriate representations | Showing no working — the method IS the evidence |
| Merit | A logical sequence of steps, connecting concepts, or forming and using a model | The “and also”: relate findings back to the context |
| Excellence | Devise a strategy, build a chain of reasoning or a proof, or form a generalisation | The “and also”: correct mathematical statements, or a stated insight |
Both Merit and Excellence are “one or more of … AND ALSO …”. A perfect answer that never interprets its result in context cannot pass the Achieved ceiling.
3 Solving trigonometric equations
General solutions, then apply any domain restriction LAST. Internal-style Solve sin(2x) = 0.5 for 0 ≤ x ≤ 2π Find every solution in the given domain. Let u = 2x. Solve sin(u) = 0.5 generally: u = π/6 + 2πn, or u = 5π/6 + 2πn. Apply the domain to u FIRST: since 0 ≤ x ≤ 2π, then 0 ≤ u ≤ 4π — twice the range, because u = 2x. List every u in [0, 4π]: π/6, 5π/6, π/6+2π=13π/6, 5π/6+2π=17π/6. NOW divide every value by 2 to return to x: π/12, 5π/12, 13π/12, 17π/12. ✓ Four solutions, not two — dividing by 2 before expanding the domain is the single most common way half the solutions get lost on this question type.4 Proving identities
One side only — never operate on both. ✕Cross-multiplying to "prove" an identity
If you move terms across the equals sign or multiply both sides by the same thing, you have assumed the two sides are already equal — which is exactly what you were asked to prove. Pick the more complicated side and simplify it in isolation until it matches the other.5 Sine and cosine rules
Choose the rule based on what you are given — and watch for the ambiguous case.Which rule, and the ambiguous case
- Two angles + any side, or two sides + a non-included angle → SINE rule.
- Two sides + the included angle, or all three sides → COSINE rule.
- SSA (two sides, non-included angle) can give TWO valid triangles — check whether the second possible angle still leaves the angle sum under 180°.
6 Fitting a sinusoidal model, end to end
The part that carries the most marks. Internal-style Tide depth peaks at 4.2 m and troughs at 0.8 m, 12.4 hours apart Build a model and use it. D = (4.2 + 0.8) ÷ 2 = 2.5 m — the midline. A = (4.2 − 0.8) ÷ 2 = 1.7 m — the amplitude. Period = 2 × 12.4 = 24.8 h (peak to trough is HALF a period), so B = 2π ÷ 24.8 ≈ 0.253. If the first high tide is at t = 3.1 h, then C = 3.1 (using the sine form shifted to peak there). Model: d = 1.7 sin(0.253(t − 3.1)) + 2.5. CONTEXT (the Merit “and also”): a boat needing 3 m clearance can cross whenever d ≥ 3 — solve the inequality and give the time window, in hours, as an answer someone could act on. ✓ Peak-to-trough is half a period — the commonest error in the whole standard. Then interpret the result in the context or it stays at Achieved.7 Reaching Excellence
Devise the strategy yourself, then generalise or evaluate the model.What an Excellence answer adds
- State the model's LIMITATIONS — over what range is it valid, and why.
- Say what would make the model fail in reality (a factor it ignores).
- Where a problem is not handed to you in standard form, devise the strategy to get there yourself.
Practice
Recall it first, then use it — testing yourself builds the memory, where rereading the notes above only feels easier.
Recall and application shuffled together — you have to work out what each item wants before you can answer it, which is the part the exam actually tests. Both kinds feed the same spacing schedule, so anything you miss comes back sooner.
Say the answer out loud before flipping — that attempt is where the learning happens, even when you get it wrong.
Exam-style only. Instant feedback, shuffled each attempt, best score saved on this device.