Grade 11 Trigonometry — How to Identify the Question Type

Grade 11 Trigonometry questions test four separate skills that look similar but need different tools: simplifying an expression using reduction formulae, proving an identity, solving an equation for a specific interval, and using the sine/cosine/area rules in a triangle that isn't right-angled. Identify which one you're looking at before picking up a formula.

Type 1: Reduction formulae

Trigger words: Angles like (180° − θ), (360° − θ), (90° + θ), or negative angles such as (−θ)

Trigger structure: An expression containing trig ratios of an angle written in terms of θ plus or minus a multiple of 90° or 180°, that must be simplified to a single ratio of θ.

Do not confuse with: Solving an equation (Type 3) — reduction questions ask you to SIMPLIFY an expression, not to find a value of θ.

Method (no numbers — just the steps)

Identify which quadrant each angle falls in (e.g. 180° − θ is in the second quadrant if θ is acute)
Apply the matching reduction formula (e.g. sin(180° − θ) = sin θ, cos(360° − θ) = cos θ)
Use the CAST diagram to assign the correct sign for that quadrant
Simplify the resulting expression to a single trig ratio of θ

Grade 11Paper 2Level 2 — Routine Procedures

Simplify, without using a calculator: sin(180° − θ) · cos(360° − θ)

Practice question — not sourced from a past paper.

Common mistake

Getting the sign wrong for the quadrant — e.g. writing cos(180° − θ) = cos θ instead of −cos θ.

See the progression — same type, increasing difficulty

Easy

Simplify: cos(180° + θ)

Practice question — not sourced from a past paper.

Medium

Simplify, without using a calculator: (sin(360° − θ))/(cos(−θ)) · tan(180° + θ)

Practice question — not sourced from a past paper.

Hard

Simplify, without using a calculator: (sin(180° − θ) · cos(90° + θ))/(sin(−θ) · cos(360° − θ))

Practice question — not sourced from a past paper.

Type 2: Proving trig identities

Trigger words: "Prove that...", "Show that... ≡ ..."

Trigger structure: An equation with trig ratios on both sides that you must show is true for all values of the variable, usually using sin²θ + cos²θ = 1 or tan θ = sin θ / cos θ.

Do not confuse with: Solving an equation (Type 3) — an identity is true for EVERY value of θ; you're not solving for a specific value.

Method (no numbers — just the steps)

Choose ONE side of the identity to start from (usually the more complicated side)
Rewrite tan θ as sin θ / cos θ wherever it appears, if useful
Use sin²θ + cos²θ = 1 to substitute or factorise where needed
Simplify step by step until that side matches the other side exactly

Grade 11Paper 2Level 3 — Complex Procedures

Prove that: (1 − cos²θ)/(sin θ) = sin θ

Practice question — not sourced from a past paper.

Common mistake

Working from both sides toward the middle at once (treating the identity like an equation to be solved) — this assumes the very thing you're trying to prove.

See the progression — same type, increasing difficulty

Easy

Prove that: sin θ · tan θ + cos θ = 1/cos θ

Practice question — not sourced from a past paper.

Medium

Prove that: (cos θ)/(1 + sin θ) + (1 + sin θ)/(cos θ) = 2/cos θ

Practice question — not sourced from a past paper.

Hard

Prove that: (sin θ − cos θ)/(sin θ + cos θ) = (1 − 2sin θ cos θ)/(1 − 2cos²θ)

Practice question — not sourced from a past paper.

Type 3: Solving trig equations within a given interval

Trigger words: "Solve for θ", with an interval given such as "0° ≤ θ ≤ 360°"

Trigger structure: An equation in θ that must be solved, restricted to a stated range of angles — at Grade 11, the general solution (with +k·360°) is NOT required.

Do not confuse with: Simplifying an expression (Type 1) — an equation has an '=' sign and asks you to find θ, not to reduce an expression to a single ratio.

Method (no numbers — just the steps)

Isolate the trig ratio on one side of the equation
Use a calculator to find the reference (acute) angle
Use the CAST diagram to find every angle within the given interval that has the same trig ratio (accounting for its sign)
List all solutions that fall inside the stated interval

Grade 11Paper 2Level 2 — Routine Procedures

Solve for θ, where 0° ≤ θ ≤ 360°: 2sin θ = 1

Practice question — not sourced from a past paper.

Common mistake

Giving only ONE solution (the calculator's reference angle) when the interval contains two or more valid solutions.

See the progression — same type, increasing difficulty

Easy

Solve for θ, where 0° ≤ θ ≤ 360°: cos θ = 0,5

Practice question — not sourced from a past paper.

Medium

Solve for θ, where 0° ≤ θ ≤ 360°: 3tan θ + 2 = −1

Practice question — not sourced from a past paper.

Hard

Solve for θ, where 0° ≤ θ ≤ 360°: 2sin²θ − sin θ − 1 = 0

Practice question — not sourced from a past paper.

Type 4: Sine, cosine and area rules

Trigger words: A triangle is described or drawn with NO right angle marked, and side lengths or angles must be found

Trigger structure: A triangle without a right angle — the standard right-angled trig ratios (SOH CAH TOA) don't apply directly.

Do not confuse with: A right-angled triangle, where SOH CAH TOA is faster and the sine/cosine rules aren't needed.

Method (no numbers — just the steps)

Check whether a right angle is marked — if so, use SOH CAH TOA instead
If two angles and a side, or two sides and a non-included angle, are known, use the sine rule: a/sin A = b/sin B = c/sin C
If two sides and the INCLUDED angle, or all three sides, are known, use the cosine rule: a² = b² + c² − 2bc·cos A
For area without a height given, use the area rule: Area = ½ab·sin C

Grade 11Paper 2Level 2 — Routine Procedures

In the diagram, P̂ = 67°, PQ = 3 cm and PR = 9,2 cm. Determine the length of QR.

Source: DBE Mathematics Paper 2, Grade 11, November 2018/2016, Question 1.1

Common mistake

Using the sine rule when two sides and the angle BETWEEN them are given — that combination needs the cosine rule, since the sine rule needs an angle opposite a known side.

See the progression — same type, increasing difficulty

Easy

In triangle PQR, angle P = 40°, angle Q = 75°, and PQ = 12 cm. Calculate the length of QR.

Practice question — not sourced from a past paper.

Medium

In triangle ABC, AB = 9 cm, AC = 7 cm, and angle BAC = 50°. Calculate the area of triangle ABC.

Practice question — not sourced from a past paper.

Hard

In triangle ABC, AB = 6 cm, BC = 8 cm, and AC = 11 cm. Calculate the size of angle ABC, correct to ONE decimal place.

Practice question — not sourced from a past paper.

Words like determine and hence appear throughout this topic — see the instruction word glossary for full definitions.