Grade 12 Trigonometry: The Complete Revision Guide

Trigonometry in Grade 12 builds on what you learned in Grades 10 and 11, and it appears in Paper 2 of the NSC Mathematics exam. Many students find trig confusing because it has many interconnected components: identities, reduction formulas, compound angles, and general solutions. But it follows clear rules, and with systematic revision, it becomes one of the more manageable sections.

The Foundation: What You Need From Earlier Grades

Before tackling Grade 12 trig, you must be solid on:

• SOHCAHTOA and the definitions of sin, cos, and tan
• The CAST diagram (which functions are positive in which quadrants)
• Special angle values: sin, cos, tan of 0°, 30°, 45°, 60°, 90°
• Solving basic trig equations (e.g., sin x = 0.5 → x = 30° or 150°)

If any of these are uncertain, spend time here first. Grade 12 trig is built on this foundation, and shaky foundations mean every advanced topic is harder than it needs to be.

Reduction Formulas

Reduction formulas allow you to simplify trig ratios of angles like (180° ± θ), (360° ± θ), and (90° ± θ) by expressing them in terms of the basic angle θ.

The most important rule: determine whether the expression is positive or negative using the CAST diagram, and then write the simplified form.

Key patterns:

• sin(180° - θ) = sin θ
• cos(180° - θ) = -cos θ
• sin(360° - θ) = -sin θ
• cos(90° - θ) = sin θ

The DBE tests these in simplification questions where you must reduce an expression to a simpler form. Practise these until the CAST diagram is instinct rather than something you have to reconstruct each time.

Trig Identities

The three Pythagorean identities are the most important:

• sin²θ + cos²θ = 1 (and its rearrangements: sin²θ = 1 - cos²θ, cos²θ = 1 - sin²θ)
• tanθ = sinθ / cosθ

Identity proofs require you to manipulate one side of an equation until it equals the other. The most common mistake is working on both sides simultaneously — you must work on one side only. Start with the more complex side and simplify toward the simpler one.

Strategy for identity proofs:

1. Express everything in terms of sin and cos
2. Factorise where possible
3. Use sin²θ + cos²θ = 1 substitutions
4. Simplify step by step

Compound Angle Formulas

Compound angle formulas are given on the information sheet in the NSC exam, which means you do not need to memorise them — but you must know how to use them:

• sin(A ± B) = sin A cos B ± cos A sin B
• cos(A ± B) = cos A cos B ∓ sin A sin B

Double angle formulas (derived from the above):

• sin 2A = 2 sin A cos A
• cos 2A = cos²A - sin²A = 1 - 2sin²A = 2cos²A - 1

Know which form of cos 2A to use in a given context — often the question or the other terms present will guide you.

General Solutions

A general solution gives all angles that satisfy a trig equation, not just those in [0°; 360°]. The format is:

For sin θ = k (where -1 ≤ k ≤ 1):

• θ = arcsin(k) + n·360° or θ = 180° - arcsin(k) + n·360° (n ∈ Z)

For cos θ = k:

• θ = ±arccos(k) + n·360°

For tan θ = k:

• θ = arctan(k) + n·180°

General solutions appear on most NSC papers. Practise writing them correctly in the required format and verify your answers by substituting specific values.

Two-Dimensional Problems

Trig questions often include triangles where you must find unknown sides or angles using the sine rule, cosine rule, or area formula. Know:

• Sine rule: a/sin A = b/sin B = c/sin C
• Cosine rule: a² = b² + c² - 2bc cos A
• Area: Area = ½ab sin C

These appear as the last question type in Paper 2 and carry 10–15 marks. Practise identifying which rule to use based on what information is given.

Access Grade 12 trigonometry past paper questions with full solutions at Examslayers Past Papers.