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Ratios and Graphs of Trigonometric Functions · Form 5

Ratios and Graphs of Trigonometric Functions: Practice Questions

Original SPM-style practice questions for Ratios and Graphs of Trigonometric Functions, each with full worked solutions, Paper 1 multiple-choice and Paper 2 structured.

Written by the spmmath.com.my editorial team.

Original practice questions for Ratios and Graphs of Trigonometric Functions, in the style of Mathematics Paper 1 (Objective) and Mathematics Paper 2 (Subjective). Each answer is worked so you can check your method, not just the result.

Multiple-choice (Paper 1 style)

Question 1

Given that sin θ < 0 and tan θ > 0 for 0° ≤ θ ≤ 360°, in which quadrant does θ lie?

  1. A. Quadrant I
  2. B. Quadrant II
  3. C. Quadrant III
  4. D. Quadrant IV

Question 2

Given cos θ = -0.6 and 90° < θ < 180°, find the value of sin θ.

  1. A. 0.8
  2. B. -0.8
  3. C. 0.6
  4. D. -0.6

Question 3

What is the reference angle (basic angle) of 210°?

  1. A. 30°
  2. B. 60°
  3. C. 150°
  4. D. 240°

Question 4

The exact value of cos 150° is

  1. A. √3/2
  2. B. -√3/2
  3. C. ½
  4. D. -½

Question 5

For 0° ≤ θ ≤ 360°, the solution set of sin θ = 0.5 is

  1. A. {30°, 150°}
  2. B. {30°, 210°}
  3. C. {150°, 330°}
  4. D. {30°, 330°}

Question 6

The value of tan 225° is

  1. A. 1
  2. B. -1
  3. C. √3
  4. D. -√3

Question 7

For the graph of y = sin x where 0° ≤ x ≤ 360°, the value of x at which y is maximum is

  1. A. 0°
  2. B. 90°
  3. C. 180°
  4. D. 270°

Question 8

From a point on level ground 15 m from the foot of a vertical tree, the angle of elevation of the top of the tree is 40°. The height of the tree is

  1. A. 12.6 m
  2. B. 11.5 m
  3. C. 9.6 m
  4. D. 17.9 m

Question 9

For 0° ≤ x ≤ 360°, how many times does the graph of y = cos x cut the x-axis?

  1. A. 1
  2. B. 2
  3. C. 3
  4. D. 4

Structured (Paper 2 style)

Question 1 (4 marks)

Given that tan θ = -5/12 and 90° ≤ θ ≤ 180°, find the values of sin θ and cos θ without using a calculator.

  1. Consider the magnitude: tan (basic angle) = 5/12, so the opposite side = 5 and the adjacent side = 12.
  2. Hypotenuse = √(5² + 12²) = √(25 + 144) = √169 = 13.
  3. θ is in Quadrant II (90° to 180°), where sine is positive and cosine is negative.
  4. Therefore sin θ = 5/13 and cos θ = -12/13.

Question 2 (3 marks)

Solve the equation cos x = -0.5 for 0° ≤ x ≤ 360°.

  1. Find the basic angle: cos⁻¹(0.5) = 60°.
  2. Cosine is negative in Quadrant II and Quadrant III.
  3. Quadrant II: x = 180° - 60° = 120°.
  4. Quadrant III: x = 180° + 60° = 240°.

Question 3 (6 marks)

A vertical tower TB stands on horizontal ground, with B at its foot. From a point P on the ground, the angle of elevation of the top T is 28° and PB = 60 m.

(a) Calculate the height of the tower TB. (b) A point Q lies on the ground between P and B.

The angle of depression of Q from T is 50°. Calculate the distance BQ.

  1. (a) In right-angled triangle TBP, tan 28° = TB ÷ PB.
  2. TB = 60 × tan 28° = 60 × 0.5317 = 31.9 m.
  3. (b) The angle of depression of Q from T equals the angle of elevation of T from Q, so this angle = 50°.
  4. In right-angled triangle TBQ, tan 50° = TB ÷ BQ, so BQ = TB ÷ tan 50° = 31.9 ÷ 1.1918 = 26.8 m.

Question 4 (6 marks)

Given that sin θ = 3/5 and θ is an acute angle, find without using a calculator: (a) cos θ, (b) tan θ, (c) sin(180° - θ), (d) cos(180° + θ).

  1. Draw a right-angled triangle: opposite = 3, hypotenuse = 5, so adjacent = √(5² - 3²) = √16 = 4.
  2. (a) cos θ = adjacent ÷ hypotenuse = 4/5.
  3. (b) tan θ = opposite ÷ adjacent = 3/4.
  4. (c) 180° - θ is in Quadrant II where sine is positive, so sin(180° - θ) = sin θ = 3/5.
  5. (d) 180° + θ is in Quadrant III where cosine is negative, so cos(180° + θ) = -cos θ = -4/5.

Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)

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Frequently asked questions

What does this trigonometric ratios and graphs practice set include?

It includes Paper 1 questions on finding trig ratios of given angles and Paper 2 structured questions on sketching sine, cosine and tangent graphs and solving trig equations from them. Every question is original practice written for this site, not a real SPM paper.

Why should I try each graph question before checking the answer?

Sketching the curve yourself, even roughly, forces you to think through the shape, period and key points of the graph. Checking the worked solution too soon means you copy the shape without understanding why it looks that way, which won't help you in the actual exam.

What's a common pitfall with trig graph questions?

A frequent error is mislabelling the axes or using the wrong scale, which throws off every reading taken from the graph. Plot enough key points before joining them into a smooth curve, and always check your angle unit against what the question asks for.

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