Grade 11 Maths · Trigonometry

Write the general solution

Write all answers to a trig equation by adding full turns, for example sin x = 0,5 gives x = 30° + k·360° or 150° + k·360°.

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General and Specific Solution grade 11:Part 1Kevinmathscience

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Question 5.1 · March 2025 · Provincial · 5 marks

5.1 Determine the general solution

cos 2 θ = - \frac{1}{2}

(5)

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More past paper questions: write the general solution

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3. Question 4.3.1 · March 2023 · Paper 1 · Provincial · 6 marks
4.3 Consider the equation: $3 cos^{2} α - sin α cos α = 0$
4.3.1 Without using a calculator, determine the general solution of this equation.(6)
Show the memo (4 ticks)Memo
1. ✓
  Factorise $cos α (3 cos α - sin α) = 0$
  1 mark · Take out a common factor
2. ✓
  Solve first factor $cos α = 0$
  $α = 9 0^{\circ} + k \cdot 18 0^{\circ}, k \in Z$
  2 marks · Write the general solution
3. ✓
  Solve second factor $3 cos α - sin α = 0$
  $sin α = 3 cos α$
  $tan α = 3$
  2 marks · Write the general solution
4. ✓
  State integer constraint $k \in Z$
  1 mark · Write the general solution
4. Question 7.2 · March 2021 · Provincial · 5 marks
7.2 Determine the general solution of: $tan 3 x \cdot \frac{1}{t a n 2 4 ^{\circ}} - 1 = 0$ (5)
Show the memo (5 ticks)Memo
1. ✓
  Isolate tan 3x $tan 3 x = tan 2 4^{\circ}$
  1 mark · Solve a linear equation
2. ✓
  Reference angle: $3 x = 2 4^{\circ}$ $3 x = 2 4^{\circ}$
  1 mark · Write the general solution
3. ✓
  Add the period $k \cdot 18 0^{\circ}$ $3 x = 2 4^{\circ} + k \cdot 18 0^{\circ}$
  1 mark · Write the general solution
4. ✓
  State $k \in Z$ $k \in Z$
  1 mark · Write the general solution
5. ✓
  Divide by 3 $x = 8^{\circ} + k \cdot 6 0^{\circ}$ , $k \in Z$
  1 mark · Write the general solution
5. Question 6.1 · November 2024 · Paper 2 · Provincial · 5 marks
6.1 Solve the following equation for $x \in [- 180°; 180°]$ .

$cos (x - 30°) = sin x$ (5)
Show the memo (5 ticks)Memo
1. ✓
  $sin x = cos (90° - x)$ $cos (x - 30°) = sin x$
  $cos (x - 30°) = cos (90° - x)$
  1 mark · Use co-ratio identities
2. ✓
  Equation with $k .360°$ $x - 30° = 90° - x + k .360°$ ; $k \in Z$
  1 mark · Write the general solution
3. ✓
  $x = 60° + k .180°$ $2 x = 120° + k .360°$
  $x = 60° + k .180°$
  1 mark · Write the general solution
4. ✓
  $k \in Z$ $k \in Z$
  1 mark · Write the general solution
5. ✓
  Solutions in the interval $x \in {- 120°; 60°}$
  1 mark · Find solutions in an interval
6. Question 4.3 · March 2022 · Provincial · 6 marks
4.3 Given $cos (x + 4 2^{\circ}) = sin 2 x$ . Solve for $x$ if $x \in [- 18 0^{\circ}; 18 0^{\circ}]$ (6)
Show the memo (5 ticks)Memo
1. ✓
  Use co-function reduction $cos (x + 4 2^{\circ}) = cos (9 0^{\circ} - 2 x)$
  1 mark · Use co-ratio identities
2. ✓
  Set up general solution equations $x + 4 2^{\circ} = 9 0^{\circ} - 2 x + k \cdot 36 0^{\circ}$ or $x + 4 2^{\circ} = - (9 0^{\circ} - 2 x) + k \cdot 36 0^{\circ}$
  2 marks · Write the general solution
3. ✓
  Solve first branch $3 x = 4 8^{\circ} + k \cdot 36 0^{\circ}$
  $x = 1 6^{\circ} + k \cdot 12 0^{\circ}$
  1 mark · Write the general solution
4. ✓
  Solve second branch $x + 4 2^{\circ} = - 9 0^{\circ} + 2 x + k \cdot 36 0^{\circ}$
  $- x = - 13 2^{\circ} + k \cdot 36 0^{\circ}$
  $x = 13 2^{\circ} - k \cdot 36 0^{\circ}$
  1 mark · Write the general solution
5. ✓
  Determine values in specific domainFor $x = 1 6^{\circ} + k \cdot 12 0^{\circ}$ : $1 6^{\circ}$ , $13 6^{\circ}$ , $- 10 4^{\circ}$
  For $x = 13 2^{\circ} - k \cdot 36 0^{\circ}$ : $13 2^{\circ}$
  $x \in {- 10 4^{\circ}; 1 6^{\circ}; 13 2^{\circ}; 13 6^{\circ}}$
  1 mark · Find solutions in an interval

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