Grade 11 Maths · Trigonometry

Use co-ratio identities

Swap between sin and cos using 90° angles, for example sin(90° − x) = cos x and cos(90° + x) = −sin x.

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Grade 11 Trigonometry Reduction with Co-FunctionsMiss Martins Maths and Science

Trigonometry - Co-Functions (Grade 11 Maths)MathsG

Cofunction Identities Examples & Practice Problems TrigonometryThe Organic Chemistry Tutor

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Question 5.2 · May/June 2025 · Paper 2 · National (DBE) · 6 marks

5.2 Simplify the expression to a single trigonometric term:

\frac{2 c o s ( 18 0 ^{\circ} - x ) s i n ( - x )}{1 - 2 c o s ^{2} ( 9 0 ^{\circ} - x )}

(6)

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More past paper questions: use co-ratio identities

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3. Question 7.1.1 · March 2024 · Paper 1 · Provincial · 3 marks
7.1 Given: $cos (A - B) = cos A cos B + sin A sin B$
7.1.1 Use the above identity to deduce that $sin (A + B) = sin A cos B + cos A sin B$ .(3)
Show the memo (3 ticks)Memo
1. ✓
  Use co-ratios to convert sine to cosine $sin (A + B) = cos [9 0^{\circ} - (A + B)]$
  $= cos [(9 0^{\circ} - A) - B]$
  1 mark · Use co-ratio identities
2. ✓
  Apply given cosine difference identity $cos (9 0^{\circ} - A) cos B + sin (9 0^{\circ} - A) sin B$
  1 mark · Use a compound angle identity
3. ✓
  Convert back using co-ratios $sin A cos B + cos A sin B$
  1 mark · Use co-ratio identities
4. Question 2.1.3 · June 2024 · Paper 2 · Provincial · 3 marks
2.1 Given: $sin 2 0^{\circ} = p$ . Determine the following in terms of $p$ :
2.1.3 $1 - sin^{2} 7 0^{\circ}$ (3)
Show the memo (3 ticks)Memo
1. ✓
  Use identity $1 - sin^{2} 7 0^{\circ} = cos^{2} 7 0^{\circ}$
  1 mark · Use the square identity
2. ✓
  Co-function $cos^{2} 7 0^{\circ} = sin^{2} (9 0^{\circ} - 7 0^{\circ}) = sin^{2} 2 0^{\circ}$
  1 mark · Use co-ratio identities
3. ✓
  Substitute p $p^{2}$
  1 mark · Use co-ratio identities
5. Question 6.3 · November 2024 · Paper 2 · Provincial · 2 marks
The graph of $f (x) = cos (x + 30°)$ in the interval of $x \in [- 120°; 150°]$ has been drawn in the diagram below. $Diagram for this question$
6.3 Write down the equation of $h$ in its simplest form.(2)
Show the memo (2 ticks)Memo
1. ✓
  $cos (90° + x)$ $h (x) = cos (x + 60° + 30°) + 1 = cos (90° + x) + 1$
  1 mark · Shift a graph
2. ✓
  $- sin x$ $h (x) = - sin x + 1$
  1 mark · Use co-ratio identities
6. Question 3.5 · June 2024 · Paper 2 · Provincial · 2 marks
In the diagram below, the graphs of $f (x) = a cos x$ and $g (x) = sin (x + p)$ are drawn for $x \in [- 18 0^{\circ}; 18 0^{\circ}]$ .
3.5 If $k (x) = cos (2 x + t)$ , determine the value of $t$ if $k (x) = g (2 x)$ .(2)
Show the memo (2 ticks)Memo
1. ✓
  Write g(2x) as a cosine $g (2 x) = sin (2 x - 3 0^{\circ}) = cos (9 0^{\circ} - (2 x - 3 0^{\circ})) = cos (12 0^{\circ} - 2 x)$
  1 mark · Use co-ratio identities
2. ✓
  Value of t $cos (12 0^{\circ} - 2 x) = cos (2 x - 12 0^{\circ})$ , therefore $t = - 12 0^{\circ}$
  1 mark · Use a reduction formula

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