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Trigonometry

Identities, reduction & equations

sin²θ + cos²θ = 1, reduction formulae, and solving trig equations in [0°; 360°].

CAPS

Grade 11 · Mathematics · Trigonometry

This sits on Mathematics Paper 2.

  • The paper

    Paper 2 is Euclidean geometry, analytical geometry, trigonometry and statistics. Diagrams are part of the question. A reason in a geometry proof must name the theorem, not only the result.

  • Cognitive level

    CAPS spreads the marks across knowledge, routine procedures, complex procedures and problem solving. A routine question still wants the line of working. A problem-solving question wants you to choose the method, not only to calculate.

  • The relation

    sin²θ + cos²θ = 1. Pythagorean identity

  • How a marker reads it

    KZN papers dock marks if you omit the second quadrant solution. Always CAST.

Each Mathematics paper is 150 marks. A correct substitution can earn a method mark even when the last arithmetic slips, if the formula was the right one.

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Diagram

Worked example

Solve 2 cos θ = 1 for θ ∈ [0°; 360°].

Before any working — hunt first

Given

Numbers and units already in the sentence: 2 cos · 1 for · 0° · 360°. Copy each one into a column. A name without a number is not given.

Missing

After you copy the given, the letter that still has no number is missing. The paper is pointing at: Solve 2 cos θ = 1 for θ ∈ [0°; 360°]. Circle that word — that is the missing variable.

Find

We are required to find: Solve 2 cos θ = 1 for θ ∈ [0°; 360°]. Write that as a symbol on the left, with a question mark, before you touch a formula.

Calculate

Write sin²θ + cos²θ = 1 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Solve 2 cos θ = 1 for θ ∈ [0°; 360°].

Simplified approach

Isolate cos θ = 12. Reference angle 60°. Cosine is positive in Q1 and Q4.

Culture note

CAST diagram is the same compass used to describe a Durban harbour tide clock — Q1 east, Q4 south-east positive cosine.

Formulas

sin²θ+cos²θ=1

Pythagorean identity

How to hunt the missing letter

Do not start by naming the letters. First hunt the missing one: copy every number you can see (Given). The letter with no number is Missing. The asking word is Find. The formula that still contains that letter is what you Calculate.

How to hunt: (1) Underline every number and its unit — those are Given. (2) Circle the asking word — find, calculate, how far, what is. That word names Find. (3) The letter the asking word names, which has no number yet, is Missing. (4) Write the formula with empty letters. Cross out every given letter. The letter still standing is what you Calculate.

  • θ

    angle

    How you spot this in the sentence

    When the sentence names the angle, that letter is θ. Copy the number next to it into Given. If the sentence never names it, θ is Missing. The unknown angle. Identities hold for every θ where they are defined.

    Example:sin² 30°+cos² 30°=(½)²+(√3/2)²=1.

  • sin θ

    sine of θ

    How you spot this in the sentence

    When the sentence names the sine of θ, that letter is sin θ. Copy the number next to it into Given. If the sentence never names it, sin θ is Missing. Opposite over hypotenuse in a right triangle; a y-coordinate on the unit circle.

    Example:sin 90°=1, sin 0°=0.

  • cos θ

    cosine of θ

    How you spot this in the sentence

    When the sentence names the cosine of θ, that letter is cos θ. Copy the number next to it into Given. If the sentence never names it, cos θ is Missing. Adjacent over hypotenuse; an x-coordinate on the unit circle.

    Example:The identity says (sin θ)² + (cos θ)² is always 1.

Exam tricks you must know

These are the moves markers look for — not shortcuts that skip the method, but the habits that earn the method mark.

  • Always CAST — the second quadrant is a mark on its own

    How:Sketch CAST. Read the sign of sin, cos or tan from the quadrant, not from memory of a special angle. For a general solution write the acute angle first, then 180° − θ or 360° − θ. Leaving the second angle is the KZN and Gauteng dock.

    When to use it:Any sin / cos / tan equation that asks for θ in an interval.

  • Memorise exact values for 0°, 30°, 45°, 60°, 90° — compound angles collapse without them

    How:Before a calculator, write the triangle: 30-60-90 and 45-45-90. sin 30° = 12, sin 45° = 1√2, sin 60° = √32, and the cosines swap. Compound-angle questions want these as fractions with division lines, not decimals.

    When to use it:Reduction, compound angles, and “without a calculator” trig.

  • Decide first: is money going in (F) or a loan being paid off (P)?

    How:Underline the asking word. “Future value”, “how much in the account”, “accumulate” → F. “Present value”, “loan”, “outstanding balance”, “how much can she borrow” → P. Write F or P in the Find cell before you open the formula sheet. The wrong letter loses the whole question.

    When to use it:Every compound-interest and annuity question on Paper 1.

Step 1 / 6

Isolate

cos θ=12.

Diagram
Diagram
cos θ=½

Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.

4. Step-by-step working

  1. 01

    Isolate

    cos θ=12.

    cos θ=½

  2. 02

    Reference

    θref=60°.

  3. 03

    Quadrants

    Q1: 60°. Q4: 360°−60°=300°.

    θ=60° or 300°

  4. 04

    Step F — Box the answer with its unit

    θ = 60° or 300°

More worked examples

Board walkthroughs on this topic. Open one, copy the working, then try the next.

Example 1 / 20

Evaluate sin 30° exactly.

4. Step-by-step working

  1. 01

    Identify the given

    sin 30° is a special ratio.

    sin

  2. 02

    Calculate

    sin 30°=12.

    12

How we got there

Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.

Explain in another language · 11

1. What is the question asking?

Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Evaluate sin 30° exactly.

Before any working — hunt first

Given

Numbers and units already in the sentence: 30°. Copy each one into a column. A name without a number is not given.

Missing

After you copy the given, the letter that still has no number is missing. The paper is pointing at: sin 30° exactly. Circle that word — that is the missing variable.

Find

We are required to find: sin 30° exactly. Write that as a symbol on the left, with a question mark, before you touch a formula.

Calculate

Write sin with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives sin 30° exactly.

In other words — as if you have never seen this before

If you have never seen this before: Memorise 0°, 30°, 45°, 60°, 90°. CAST for 180°. We will name a method in words, write the formula with no numbers, then substitute one line at a time.

2. What do we already know?

Special angle 30°, function sin.

3. Which method — and why this one?

Memorise 0°, 30°, 45°, 60°, 90°. CAST for 180°.

4. Step-by-step working

  1. 01 / 05

    Identify the given

    sin 30° is a special ratio.

    sin

    Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.

  2. 02 / 05

    Calculate

    sin 30°=12.

    12

    Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.

  3. 03 / 05

    Step G — Sense-check the size

    Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.

    Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.

  4. 04 / 05

    Step F — Box the answer with its unit

    12

    Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.

  5. 05 / 05

    Step H — What the marker is looking for

    On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.

    Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.

5. The answer

12

Copy onto your answer sheet — in this order

Formula: sin

Answer: 12

6. Check it like a marker

Do not give a rounded decimal when the paper wants an exact surd.

7. A trap first-timers fall into

First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”

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