Grade 11 Trigonometry

Grade 11 trigonometry: reduction formulae, identities, general solutions, the sine, cosine and area rules, and trig graphs with shifts, with worked examples.

Trigonometry is the biggest topic in Grade 11 Paper 2, at about 50 of the 150 marks. You prove and use the identities tan θ = sin θ / cos θ and sin²θ + cos²θ = 1, simplify expressions with the reduction formulae (180° ± θ, 360° − θ, −θ and 90° ± θ), and find the general solution of trig equations.

For triangles that are not right-angled you use the sine rule, cosine rule and area rule. You also sketch trig graphs with a period change (y = sin kx) and a horizontal shift (y = sin(x + p)). Grade 12 adds compound and double angles and 3D problems.

Key concepts

  • Identities: tan θ = sin θ / cos θ and sin²θ + cos²θ = 1
  • Reduction formulae and the CAST diagram
  • General solutions, e.g. sin θ = 0.5 ⇒ θ = 30° + k·360° or θ = 150° + k·360°
  • Sine rule: a/sin A = b/sin B = c/sin C
  • Cosine rule: a² = b² + c² − 2bc cos A
  • Area rule: Area = ½ab sin C
  • Graphs of y = sin kx, y = cos kx, y = sin(x + p) and their periods

Worked examples

Simplify: sin(180° − θ) · cos(360° − θ) ÷ tan(180° + θ)

Reduction formulae give sin θ · cos θ ÷ tan θ = sin θ · cos θ · cos θ / sin θ = cos²θ.

Find the general solution of 2 sin θ = 1.

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

In △ABC, a = 7, b = 8 and Ĉ = 60°. Find c.

Cosine rule: c² = 7² + 8² − 2(7)(8)cos 60° = 49 + 64 − 56 = 57, so c = √57 ≈ 7.55.

Find the area of △PQR with PQ = 10 cm, PR = 6 cm and P̂ = 30°.

Area = ½ × PQ × PR × sin P = ½ × 10 × 6 × ½ = 15 cm².

Common mistakes

  • Giving only one solution of a trig equation instead of the general solution.
  • Using the sine rule when you have two sides and the included angle (that needs the cosine rule).
  • Getting the sign wrong in a reduction formula (check the quadrant).
  • Leaving out k ∈ ℤ in the general solution.

Exam tips

  • For reduction, decide the quadrant first, then the sign, then whether the function changes (only for 90° ± θ).
  • Use the cosine rule for SAS and SSS; use the sine rule for AAS and SSA.
  • In identity proofs, work on one side only and change everything to sin and cos.

Frequently asked questions

When do I use the sine rule versus the cosine rule?

Use the cosine rule when you know two sides and the angle between them, or all three sides. Use the sine rule when you know a side and its opposite angle plus one more side or angle.

What is a general solution?

A formula that gives every angle that solves the equation, written with k·360° (or k·180° for tan), where k is any integer.

Are double angles in Grade 11?

No. Compound and double angle identities, such as sin 2θ = 2 sin θ cos θ, are Grade 12 work.

Snap&Learn gives South African Matric learners instant, CAPS-aligned, step-by-step Mathematics solutions. Snap or upload a question and the AI shows every method mark the way the NSC memo awards them. Your first three AI solutions are free.