Trigonometric functions — Paper 2 · IGCSE Mathematics 0580

8 Examiner's y Use 1 y = sinx 0.5 0 x 30° 60° 90° 120° 150° 180° –0.5 y = cosx –1 The diagram shows accurate graphs of y = sinx and y = cosx for 0° Y x Y 180°. Use the graph to solve the equations (a) sinx – cosx = 0, Answer(a) x = [1] (b) sinx – cosx = 0.5. Answer(b) x = [2]

3 marks

Mark scheme: 8 (a) 43 to 47 1 (b) 64 to 68 2 SC1 23 to 27

This question in 0580/22 May/June 2010

24 For Q P Examiner's Use 6 cm NOT TO SCALE C B 5 cm D 10 cm A The diagram shows a triangular prism. ABCD is a horizontal rectangle with DA = 10 cm and AB = 5 cm. BCQP is a vertical rectangle and BP = 6 cm. Calculate (a) the length of DP, Answer(a) DP = cm [3] (b) the angle between DP and the horizontal rectangle ABCD. Answer(b) [3]

6 marks

Mark scheme: 24 (a) 12.7 3 M2 for 102 + 52 + 62 or M1 for one of 102 + 52 or 62 + 52 or 102 + 62 (b) 28.2 3 M2 for sin x = 6/(a) or M1 for identifying angle PDB 70

This question in 0580/23 Oct/Nov 2012

10 Calculate. (a) 2 3 - 10 + 4 2 … [1] 2 3 # tan 70° (b) 3 … [1]

2 marks

Mark scheme: 10 (a) 2.9[0] or 2.900 to 2.901 1 (b) 3.17 or 3.172 to 3.173 1  40  10 

This question in 0580/23 Oct/Nov 2016

12 x° is an obtuse angle and sin x° = 0.43 . Find the value of x. x = … [2]

2 marks

Mark scheme: 12 154.5 or 154.5… 2 B1 for 25.5 or 25.46 to 25.47 or M1 for 180 – sin–1(0.43) oe

This question in 0580/22 Oct/Nov 2018

8 When sin x° = 0.36, find (a) the acute angle x°, … [1] (b) the obtuse angle x°. … [1]

2 marks

Mark scheme: 8(a) 21.1 or 21.10… 1 8(b) 158.9 or 158.8 to 158.9 1 FT 180 – their (a) providing answer is an obtuse angle

This question in 0580/23 Oct/Nov 2019

19 y 1 0 x 90° 180° 270° 360° – 1 (a) On the diagram, sketch the graph of y = cos x for 0° G x G 360° . [2] (b) Solve the equation 4 cosx + 2 = 3 for 0° G x G 360 ° . x = … and x = … [3] Questions 20 and 21 are printed on the next page.

5 marks

Mark scheme: 19(a) Correct sketch 2 Needs all three features for 2 marks: • Correct curve shape • Maximum at (0, 1) and at (360, 1) and minimum at (180, −1) • Passing through (90, 0) and (270, 0) only B1 for two correct features 19(b) 75.5 or 75.52… 3 B2 for one correct and 1 or M1 for cos x = oe 284.4 to 284.5 4 If 0 scored, SC1 for two answers with a sum of 360

This question in 0580/22 Feb/March 2020

25 Solve the equation tanx = 2 for 0° G x G 360° . x = … or x = … [2]

2 marks

Mark scheme: 25 63.4 or 63.43... 2 B1 for each 243.4 or 243.4... If 0 scored SC1 for two answers with a difference of 180

This question in 0580/22 Oct/Nov 2020

25 Solve 3 tanx =- 4 for 0° G x G 360° . x = … or x = … [3]

3 marks

Mark scheme: 25 126.9 or 126.86 to 126.87 and 306.9 or 3 B2 for one correct 306.86 to 306.87 4 or M1 for tan x = − 3 if 0 scored then SC1 for two answers with a difference of 180°

This question in 0580/23 Oct/Nov 2020

23 Find all the solutions of 4 sinx = 3 for 0° G x G 360 ° . … [2]

2 marks

Mark scheme: 23 48.6 or 48.59… 2 B1 for each and If 0 scored SC1 for two answers with a sum of 180° 131.4 or 131.4…

This question in 0580/22 May/June 2021

20 Solve 3 ( 2 + cosx) = 5 for 0° G x G 360° . … [3]

3 marks

Mark scheme: 20 109.4 to 109.5 3 B2 for one correct angle and 250.5 to 250.6 5 or M1 for cos x = − 2 or better 3 If 0 scored SC1 for two angles that sum to 360

This question in 0580/21 Oct/Nov 2021

22 Solve the equation 7 sinx + 2 = 0 for 0° G x G 360° . … [3] Question 23 is printed on the next page.

3 marks

Mark scheme: 22 196.6 or 196.60… 3 B2 for one correct angle and 343.4 or 343.39… 2 or M1 for sin x = − or better 7 If 0 scored SC1 for two angles that sum to 540°

This question in 0580/22 Oct/Nov 2021

24 tanx = 3 and 0° G x G 360° . Find all the possible values of x. … [2]

2 marks

Mark scheme: 24 60 2 B1 for 60 or 240 and 240 If 0 scored SC1 for two answers with a difference of 180°

This question in 0580/23 Oct/Nov 2021

17 (a) Sketch the graph of y = sin x for 0° G x G 360° . y 1 O x 360 – 1 [2] (b) Solve the equation 3 sinx + 1 = 0 for 0° G x G 360° . x = … or x = … [3]

5 marks

Mark scheme: 17(a) 2 Correct sketch to go through B1 for correct sine curve shape through the origin (0, 0), (180, 0) and (360, 0) 17(b) 199.5 or 199.47… 3 B2 for one correct and 340.5 or 340.52 to 340.53… 1 or M1 for sin x =  oe 3 If 0 scored SC1 for two reflex angles with sum of 540 or two non-reflex angles with sum of 180

This question in 0580/22 May/June 2022

23 Solve the equation 3 sinx + 3 = 1 for 0° G x G 360° . x = … or x = … [3]

3 marks

Mark scheme: 23 221.8 or 221.81... 3 B2 for one correct and 2 or M1 for sin x = – oe 318.2 or 318.18 to 318.19 3 If 0 scored, SC1 for two reflex angles with a sum of 540 or two non-reflex angles with a sum of 180

This question in 0580/23 May/June 2022

21 Solve the equation 5 sinx = - 3 for 0° G x G 360° . … [3] Questions 22 and 23 are printed on the next page.

3 marks

Mark scheme: 21 216.9 or 216.86 to 216.87 3 323.1 or 323.13… B2 for one correct angle 3 or M1 for sinx =  or better 5 If M1 or 0 scored SC1 for two reflex angles with a sum of 540 or two non-reflex angles with a sum of 180

This question in 0580/22 May/June 2023

19 (a) y 1 x 0 360° – 1 Sketch the graph of y = cos x for 0° G x G 360 ° . [2] (b) When cosx = 0 .21 , find the reflex angle x. … [2]

4 marks

Mark scheme: 19(a) 2 B1 for correct cosine curve shape through (0,1) Correct sketch to go through (0, 1), close to (360, 1) and reasonably close to (180, –1) 19(b) 282.1 or 282.12… 2 B1 implied by 77.9 or 77.87 to 77.88 or 282.13 or M1 for 360 – their acute angle

This question in 0580/21 Oct/Nov 2023

20 Solve the equation 8 sinx + 6 = 1 for 0c G x G 360 c . x = … or x = … [3]

3 marks

Mark scheme: 20 218.7, 321.3 3 B2 for one correct 5 or M1 for sin x = – oe 8 If M1 or 0 scored, SC1 for two reflex angles with a sum of 540 or two non-reflex angles with a sum of 180

This question in 0580/21 May/June 2024

22 Solve the equation tanx + 3 = 0 for 0° G x G 360° . … [3]

3 marks

Mark scheme: 22 120, 300 3 B2 for one correct or M1 for tan x =  3 oe If 0 or M1 scored SC1 for answers with difference of 180

This question in 0580/23 May/June 2024

21 Solve the equation 3 tanx + 5 = 1 for 0° G x G 360° . x = … or x = … [3]

3 marks

Mark scheme: 21 126.9 and 306.9 3 B2 for one correct answer 4 or M1 for tan x = – oe 3 If M1 or 0 scored, SC1 for two angles with a difference of 180

This question in 0580/21 Oct/Nov 2024

22 (a) Write down the exact value of tan 60°. … [1] (b) Solve 2 sinx - 1 = 0 for 0 ° G x G 360° . x = … or x = … [3]

4 marks

Mark scheme: 22(a) 3 1 22(b) 30, 150 3 B2 for 30 or 150 1 or M1 for sin x = 2 If 0 or M1 scored, SC1 for one acute angle and one obtuse angle with a sum of 180

This question in 0580/22 May/June 2025

1 26 Solve tanx =- for 0° G x G 360° . 3 x = … , x = … [3]

3 marks

Mark scheme: 26 150 and 330 3 B2 for one correct answer or B1 for [tan–1…] = 30 or –30 If 0 scored, SC1 for two answers in range with a difference of 180

This question in 0580/21 Oct/Nov 2025

21 (a) y 1 O 180° 360° x – 1 On the diagram, sketch the graph of y = cos x for 0° G x G 360° . [2] (b) Solve the equation 2 cosx + 3 = 0 for 0° G x G 360° . x = … or x = … [3]

5 marks

Mark scheme: 21(a) Correct sketch to go through (0, 1), 2 M1 for correct cos curve shape through (180, –1) and (360, 1) (0, 1) or for almost correct sketch within 1 tolerance but with an omission at either end 0 or for almost correct sketch within 360 tolerance but with incorrect curvature in one place only –1 21(b) 150 and 210 3 B2 for one correct answer 3 or M1 for cos x = − or better 2 or B1 for 30 or –30 If M1, B1 or 0 scored, award SC1 for two answers in range with a sum of 360

This question in 0580/22 Oct/Nov 2025

17 (a) Sketch the graph of y = cos x for 0 ° G x G 360 ° . y 1 0 x 180° 360° –1 [2] 1 (b) cos x° = and x is a reflex angle. 2 Find the value of x. x = … [2]

4 marks

Mark scheme: 17(a) Correct sketch to go through (0, 1), 2 M1 for correct cos curve shape through (180, –1) and (360, 1) (0, 1) or for almost correct sketch within 1 tolerance but with an omission at either end 0 or for almost correct sketch within 360 tolerance but with incorrect curvature in one place only –1 17(b) 315 2 B1 for 45

This question in 0580/23 Oct/Nov 2025