Grade 10 Maths · Trigonometry

Decide the quadrant and sign

Decide which quadrant an angle lies in and whether x, y or the ratio is positive or negative, for example 150° is in the second quadrant, so cos is negative.

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Question 4.2 · March 2025 · Provincial · 3 marks

4.2 If

tan α = \frac{3}{4}

where

9 0^{\circ} < α < 27 0^{\circ}

. Determine, with the aid of a diagram, the value of

cos α

.(3)

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More past paper questions: decide the quadrant and sign

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3. Question 2.3.1 · June 2024 · Paper 2 · Provincial · 4 marks
2.3 Given: $k sin β = - 5$ and $k cos β - 1 = 0$ where $k > 0$ .
2.3.1 Explain why $β \in [27 0^{\circ}; 36 0^{\circ}]$ (4)
Show the memo (4 ticks)Memo
1. ✓
  Isolate sin beta $sin β = - \frac{5}{k}$
  1 mark · Solve a linear equation
2. ✓
  Isolate cos beta $cos β = \frac{1}{k}$
  1 mark · Solve a linear equation
3. ✓
  DiagramSketch: terminal arm through the point $(1; - 5)$ with $r = k$ , in quadrant IV (see diagram).
  1 mark · Decide the quadrant and sign
4. ✓
  ExplanationSince $k > 0$ : $sin β$ is negative and $cos β$ is positive, which happens only in quadrant IV, so $β \in [27 0^{\circ}; 36 0^{\circ}]$ . OR In quadrant IV, $y$ is negative and $x$ is positive.
  1 mark · Decide the quadrant and sign
4. Question 3.1.1 · March 2023 · Paper 1 · Provincial · 2 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:
3.1.1 The value of x.(2)
Show the memo (2 ticks)Memo
1. ✓
  Use Pythagoras theorem $x^{2} + y^{2} = r^{2}$
  $x^{2} + (- 9)^{2} = 1 5^{2}$
  $x^{2} = 225 - 81 = 144$
  1 mark · Use the theorem of Pythagoras
2. ✓
  Determine sign based on quadrantP is in 3rd quadrant (diagram shows x < 0).
  $x = - 12$
  1 mark · Decide the quadrant and sign
5. Question 3.2.1 · June 2024 · Paper 2 · Provincial · 3 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:
3.2.1 $cos β$ (3)
Show the memo (3 ticks)Memo
1. ✓
  Diagram (quadrant and signs of x and y) $tan β = - \frac{2}{3}$ and $sin β > 0$ , so $β$ is in the 2nd quadrant: sketch with $x = - 3$ , $y = 2$
  1 mark · Decide the quadrant and sign
2. ✓
  Value of r $r^{2} = (- 3)^{2} + (2)^{2}$ $r = 13$
  1 mark · Use the theorem of Pythagoras
3. ✓
  Calculate cos β $cos β = \frac{x}{r}$
  $cos β = - \frac{3}{13}$
  1 mark · Find trig ratios from a sketch
6. Question 2.3.1 · March 2022 · Provincial · 3 marks
2.3 If $5 cos A + 3 = 0$ and $A \in [0^{\circ}; 18 0^{\circ}]$ , determine WITHOUT the use of calculator:
2.3.1 $sin A$ (3)
Show the memo (3 ticks)Memo
1. ✓
  Determine cos A and quadrant $cos A = - \frac{3}{5}$
  Since $A \in [0^{\circ}; 18 0^{\circ}]$ and $cos A$ is negative, A is in Quadrant 2.
  1 mark · Decide the quadrant and sign
2. ✓
  Calculate y $x = - 3, r = 5$
  $x^{2} + y^{2} = r^{2} ⟹ (- 3)^{2} + y^{2} = 5^{2}$
  $y^{2} = 16 ⟹ y = 4$ (positive in Q2)
  1 mark · Use the theorem of Pythagoras
3. ✓
  Determine sin A $sin A = \frac{y}{r} = \frac{4}{5}$
  1 mark · Find trig ratios from a sketch

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