Trigonometry for Matric Maths

Key concepts

• Compound and double angle identities
• Reduction formulae (90° ± θ, 180° ± θ, 360° − θ)
• General solutions of trig equations
• Sine, cosine and area rules in 2D and 3D
• Sketching y = sin x, y = cos x, y = tan x with transformations
• tan θ = sin θ / cos θ and sin²θ + cos²θ = 1
• Negative angles: sin(−θ) = −sin θ and cos(−θ) = cos θ
• Special angles 30°, 45° and 60° without a calculator
• Proving identities by working on one side at a time

Worked examples

Simplify sin(180° − x) · cos(360° − x) / cos(90° − x)

Use reductions: sin(180° − x) = sin x, cos(360° − x) = cos x, cos(90° − x) = sin x. Expression = (sin x · cos x) / sin x = cos x.

Find the general solution of 2 sin x = 1.

sin x = ½, so the reference angle is 30°. Sine is positive in quadrants 1 and 2: x = 30° + k·360° or x = 150° + k·360°, k ∈ ℤ.

Without a calculator, find cos 75°.

cos 75° = cos(45° + 30°) = cos 45° cos 30° − sin 45° sin 30° = (√2/2)(√3/2) − (√2/2)(½) = (√6 − √2)/4.

Prove that (1 − cos 2x) / sin 2x = tan x.

LHS = (1 − (1 − 2sin²x)) / (2 sin x cos x) = 2sin²x / (2 sin x cos x) = sin x / cos x = tan x = RHS.

In △ABC, a = 7, b = 5 and C = 60°. Find c.

Cosine rule: c² = 7² + 5² − 2(7)(5)cos 60° = 49 + 25 − 35 = 39, so c = √39 ≈ 6.24.

Common mistakes

• Forgetting the +k·360° (or k·180°) when writing general solutions.
• Mixing up sine rule and cosine rule conditions.
• Working in degree mode when the question is in radians (or vice-versa).
• Dividing both sides of an equation by sin x or cos x, which loses solutions.
• Working on both sides of an identity at once instead of showing LHS = RHS.

Exam tips

• Memorise the identities from the formula sheet — but practise applying them under time pressure.
• For 3D problems, draw and label every triangle separately.
• Show the quadrant reasoning for each reduction so the marker can follow your signs.
• In 3D problems, identify which triangles are right-angled before choosing a rule.

In past papers

• NSC Nov 2024 Paper 2 Q5–Q7 — identities, equations and 3D triangle

Frequently asked questions

Which identities are on the CAPS formula sheet?

Compound and double angle identities are provided. You still need to know the basic Pythagorean identities and how to apply reductions.

How do I write a general solution?

Find the reference angle, use the CAST diagram to find the quadrants where the ratio has the correct sign, then add k·360° for sine and cosine or k·180° for tangent, with k ∈ ℤ.

Are trig graphs in Paper 1 or Paper 2?

Paper 2. You sketch and interpret graphs such as y = a sin bx, y = cos(x + p) and y = tan x, and read solutions or inequalities from them.