Grade 10 Maths · Trigonometry

Use special angle values

Give exact trig values for 0°, 30°, 45°, 60° and 90° without a calculator, for example sin 30° = ½ and tan 45° = 1.

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Question 3.3 · March 2025 · Provincial · 5 marks

In the diagram,

△ PQR

is a right-angled triangle,

\hat{Q} = 9 0^{\circ}

\hat{P} = θ

PQ = 5

and

PR = 13

units.

3.3 Without using a calculator, determine the value of:

\frac{cosec 4 5 ^{\circ} \cdot s i n 6 0 ^{\circ}}{s i n 4 5 ^{\circ} \cdot c o s 3 0 ^{\circ}}

(5)

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More past paper questions: use special angle values

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3. Question 2.3 · June 2024 · Paper 2 · Provincial · 3 marks
2.3 Evaluate without using a calculator: $\frac{s i n 4 5 ^{\circ}}{\frac{2}{2} \cdot t a n ^{2} 3 0 ^{\circ}}$ (3)
Show the memo (2 ticks)Memo
1. ✓
  Substitute special angles $= \frac{\frac{1}{2}}{\frac{2}{2} \times ( \frac{1}{3} ) ^{2}}$
  2 marks · Use special angle values
2. ✓
  Simplify $= \frac{\frac{2}{2}}{\frac{2}{2} \cdot \frac{1}{3}}$
  $= \frac{1}{\frac{1}{3}}$
  $= 3$
  1 mark · Use special angle values
4. Question 3.3 · March 2024 · Provincial · 5 marks
A right angled triangle with sides $p$ , $q$ and $s$ and the angle $θ$ is given. (Based on diagram: $p$ = Hypotenuse, $q$ = Adjacent, $s$ = Opposite to $θ$ )
3.3 Without the use of the calculator, showing all your working, determine the value of: $\frac{s i n 4 5 ^{\circ} \cdot t a n ^{2} 6 0 ^{\circ}}{c o s 4 5 ^{\circ}} + sin 3 0^{\circ}$ (5)
Show the memo (3 ticks)Memo
1. ✓
  Substitute special angles $\frac{( \frac{2}{2} ) \cdot ( 3 ) ^{2}}{\frac{2}{2}} + \frac{1}{2}$
  3 marks · Use special angle values
2. ✓
  Simplify $(\frac{2}{2} \div \frac{2}{2}) \cdot 3 + \frac{1}{2}$
  $= 1 \cdot 3 + \frac{1}{2}$
  1 mark · Use special angle values
3. ✓
  Final Answer $3 \frac{1}{2}$ or $\frac{7}{2}$
  1 mark · Use special angle values
5. Question 4.3 · March 2023 · Provincial · 5 marks
4.3 Without the use of the calculator, showing all your working, determine the value of: $\frac{c o t ^{2} 6 0 ^{\circ} \cdot t a n 4 5 ^{\circ} \cdot s i n 3 0 ^{\circ}}{c o s 6 0 ^{\circ}}$ (5)
Show the memo (5 ticks)Memo
1. ✓
  cot^2 60° $cot^{2} 6 0^{\circ} = (\frac{3}{3})^{2}$
  1 mark · Use special angle values
2. ✓
  tan 45° $tan 4 5^{\circ} = 1$
  1 mark · Use special angle values
3. ✓
  sin 30° $sin 3 0^{\circ} = \frac{1}{2}$
  1 mark · Use special angle values
4. ✓
  cos 60° $cos 6 0^{\circ} = \frac{1}{2}$
  1 mark · Use special angle values
5. ✓
  Answer $\frac{( \frac{3}{3} ) ^{2} ( 1 ) ( \frac{1}{2} )}{\frac{1}{2}} = \frac{1}{3}$
  1 mark · Use special angle values
6. Question 5.2 · March 2018 · Provincial · 3 marks
5.2 Simplify the following WITHOUT the use of a calculator: $\frac{c o s 3 0 ^{\circ}}{cosec 6 0 ^{\circ}}$ (3)
Show the memo (3 ticks)Memo
1. ✓
  Substitute special angles $cos 3 0^{\circ} = \frac{3}{2}$
  1 mark · Use special angle values
2. ✓
  Substitute denominator $cosec 6 0^{\circ} = \frac{2}{3}$
  1 mark · Use special angle values
3. ✓
  Simplify fraction $\frac{\frac{3}{2}}{\frac{2}{3}} = \frac{3}{2} \times \frac{3}{2} = \frac{3}{4}$
  1 mark · Use special angle values

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