Trigonometry: The Complete Guide (~40–50 marks, Paper 2)

#

Trigonometry is worth 40–50 marks in Paper 2 — one of the heaviest topics in the entire exam. It spans identities, equations, general solutions, and 2D/3D applications.

What Makes Grade 12 Trig Different?

#

In Grade 10–11 you worked with single angles and basic identities. In Grade 12, two new powers are unlocked:

1. Compound Angle Identities: Breaking apart $\sin(\alpha + \beta)$ and $\cos(\alpha - \beta)$ into workable pieces.
2. Double Angle Identities: Finding $\sin 2\theta$ and $\cos 2\theta$ when you only know single-angle values.

These identities are the engine behind proving identities, solving equations, and finding general solutions.

The Compound Angle Formulas (Must Memorise)

#

⚠️ Memory trick: For sin, the sign in the middle matches. For cos, the sign flips.

The Double Angle Formulas

#

Set $\beta = \alpha$ in the compound angle formulas:

💡 $\cos 2\alpha$ has three forms — choosing the right one is often the key to solving a problem. If the rest of the expression has $\sin$, use $1 - 2\sin^2\alpha$. If it has $\cos$, use $2\cos^2\alpha - 1$.

General Solutions

#

Where $n \in \mathbb{Z}$ (any integer) and $\theta_{\text{ref}}$ is the reference angle from your calculator.

The Foundation You Need (Grade 11 Revision)

#

Before diving in, make sure you are solid on:

• The CAST diagram (which trig ratios are positive in each quadrant)
• Reduction formulae ($\sin(180° - \theta) = \sin\theta$, etc.)
• Special angles: $30°$, $45°$, $60°$ and their exact trig values
• Basic identities: $\tan\theta = \frac{\sin\theta}{\cos\theta}$ and $\sin^2\theta + \cos^2\theta = 1$

Deep Dives (click into each)

#

• Compound & Double Angle Identities — the core formulas that power all of Grade 12 Trig
• Double Angle Identities (Deep Dive) — the three faces of $\cos 2\theta$ and when to use each
• Proving Trigonometric Identities — the strategy and worked examples for identity proofs
• Solving Trig Equations & General Solutions — finding every angle that satisfies an equation
• 2D & 3D Trigonometry Problems — Sine Rule, Cosine Rule, area formula, and multi-plane problems

🚨 Common Mistakes

#

1. Wrong compound angle formula sign: $\cos(\alpha + \beta)$ has a minus in the middle, not plus. The sign flips compared to $\sin$.
2. Choosing the wrong $\cos 2\alpha$ form: If your expression involves $\sin^2\theta$, use $\cos 2\theta = 1 - 2\sin^2\theta$. Wrong form = dead end.
3. General solution — forgetting $n \in \mathbb{Z}$: Always state that $n$ is an integer.
4. Missing solutions in restricted domains: After finding the general solution, substitute $n = 0, \pm 1, \pm 2, \ldots$ to find ALL solutions in the given interval.
5. 3D problems — wrong triangle: In multi-plane problems, identify which triangle contains the angle/side you need. Draw it separately.
6. Not using the area rule: When a question says “find the area of triangle”, use $\text{Area} = \frac{1}{2}ab\sin C$, not $\frac{1}{2} \times \text{base} \times \text{height}$ (unless you know the height).

🔗 Related topics:

• Euclidean Geometry — geometry proofs sometimes combine with trig
• Analytical Geometry — angle of inclination uses $\tan\theta = m$

📌 Grade 11 foundation: Trigonometry — CAST, reduction, identities, general solutions, sine/cosine rules

⏮️ Probability | 🏠 Back to Grade 12 | ⏭️ Analytical Geometry