Grade 10 Maths · Trigonometry

Find trig ratios from a sketch

Find trig ratios from a point or a sketch using x, y and r, for example the point (−3; 4) gives r = 5 and cos θ = −3/5.

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Question 5.1.1 · May/June 2025 · Paper 2 · National (DBE) · 3 marks

5.1 If

cos θ = - \frac{5}{13}

where

18 0^{\circ} < θ < 36 0^{\circ}

, determine, without using a calculator, the value of:

5.1.1

sin^{2} θ

(3)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

More past paper questions: find trig ratios from a sketch

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3. Question 3.1.2 · March 2024 · Paper 1 · Provincial · 3 marks
Do not use a calculator when answering Question 3.
In the diagram, $P (- 6; 8)$ is a point in the Cartesian plane. A is a point on the positive x-axis. $∠ A OP = θ$ .
3.1 Determine, with the aid of the diagram, the following:
3.1.2 $sin θ + cos θ$ (3)
Show the memo (2 ticks)Memo
1. ✓
  Determine ratios $sin θ = \frac{y}{r} = \frac{8}{10}$
  $cos θ = \frac{x}{r} = \frac{- 6}{10}$
  2 marks · Find trig ratios from a sketch
2. ✓
  Calculate sum $\frac{8}{10} - \frac{6}{10} = \frac{2}{10} = \frac{1}{5}$
  1 mark · Find trig ratios from a sketch
4. Question 3.2.2 · June 2024 · Paper 2 · Provincial · 2 marks
3.2 If $3 tan β + 2 = 0$ and $sin β > 0$ , determine without the use of a calculator and with the aid of a sketch, the value of the following:
3.2.2 $13 cos β - 13 sin^{2} β$ (2)
Show the memo (2 ticks)Memo
1. ✓
  Substitution $13 (- \frac{3}{13}) - 13 (\frac{2}{13})^{2}$
  1 mark · Find trig ratios from a sketch
2. ✓
  Answer $= - 3 - 4 = - 7$
  1 mark · Find trig ratios from a sketch
5. Question 3.1.3 · March 2023 · Paper 1 · Provincial · 3 marks
In the diagram, $P (x; - 9)$ is a point in the Cartesian plane such that $OP = 15$ units. $S (- 2; c)$ lies on $OP$ . $R$ is a point on the positive x-axis and $R \hat{O} P = θ$ .
3.1 Determine, with the aid of the diagram, the following:
3.1.3 The value of c.(3)
Show the memo (3 ticks)Memo
1. ✓
  Use tan theta ratio for point SFor point S(-2; c): $tan θ = \frac{c}{- 2}$
  1 mark · Find trig ratios from a sketch
2. ✓
  Equate ratios $\frac{c}{- 2} = \frac{3}{4}$
  1 mark · Find trig ratios from a sketch
3. ✓
  Solve for c $4 c = - 6$
  $c = - \frac{3}{2}$ (or $- 1, 5$ )
  1 mark · Find trig ratios from a sketch
6. Question 3.2.3 · March 2023 · Provincial · 2 marks
In $Δ A BC$ , $\hat{C} = 9 0^{\circ}$ and $\hat{B} = θ$ , $A C = 7$ units, $BC = 5$ units and $A B = 74$ units.
3.2 In the diagram below, $Q (x; y)$ is a point in the second quadrant. $R \hat{O} Q = θ$ and $13 sin θ = 12$ . Make use of the diagram to:
3.2.3 Calculate the value of $5 tan θ + 1$ .(2)
Show the memo (2 ticks)Memo
1. ✓
  Substitute ratio $5 (- \frac{12}{5}) + 1$
  1 mark · Find trig ratios from a sketch
2. ✓
  Calculate $- 12 + 1 = - 11$
  1 mark · Find trig ratios from a sketch

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