TRIGONOMETRY RATIOS. LCOL and JCHL Revision

Transcription

1 TRIGONOMETRY RATIOS LCOL and JCHL Revision

2 2017 JCHL Paper 2 Question 8 (a) (i) The diagram below shows two right-angled triangles, ABC and ACD. They have right angles at B and D, respectively. AB = 10, AC = 12, and AD = DC = x, as shown. The angle BAC is marked Y. Use trigonometry to find the size of the angle Y. Give your answer correct to one decimal place. Hypotenuse 12 Y 10 Adjacent cos Y = Y = cos 1 12 Y Marks

3 2017 JCHL Paper 2 Question 8 (a) (ii) Find the value of x. Give your answer correct to two decimal places. Pythagoras c 2 = a 2 + b 2 12 x x c 2 = a 2 + b = x 2 + x = 2x 2 72 = x 2 x = 72 x = 8.49 units 10 Marks

4 2017 JCHL Paper 2 Question 12 (a) The diagram on the right shows a right-angled triangle with a of length 10 units. Use trigonometry to find the length of the side marked x. Give your answer in surd form. Hypotenuse Opposite sin 60 = x 10 x = 10 sin 60 x = 5 3 units 10 Marks

5 2017 JCHL Paper 2 Question 12 (b) The diagram below shows a regular hexagon with sides of length 10 units. The hexagon is divided into 6 equilateral triangles. Work out the area of this hexagon. Give your answer in the form a 3, where a N. Area of a Triangle = 1 2 base perpendicular height There are 6 equilateral triangles of base 10 and height 5 Area of Hexagon = = units Marks

6 2016 JCHL Paper 2 Question 4 (c) Use trigonometry to find the measure of the angle ABC. Give your answer in degrees, correct to two decimal places. Opposite 5 Let X= ABC 6 X 5 6 X Adjacent tan X = 5 6 X = tan X = ABC = Marks

7 2016 JCHL Paper 2 Question 8 (a) (i) Write in degrees in decimal form, correct to two decimal places. Enter into the calculator using the degrees, minutes, seconds button. Turn into decimal using the SD button = 2.72 (ii) Write 3 14 in DMS (i.e. degrees, minutes, and seconds) = Marks

8 2016 JCHL Paper 2 Question 8 (b) The diagram shows a right-angled triangle, with the angle A marked. Given that cos A = sin A, show that this triangle must be isosceles. cos A = sin A = = Hypotenuse Opposite The side is equal in length to the side there fore the triangle is isosceles. Adjacent 10 Marks

9 2016 JCHL Paper 2 Question 8 (c) A right-angled triangle has sides of length 7 cm, 24 cm, and 25 cm. Find the size of the smallest angle in this triangle. Give your answer correct to one decimal place. X Adjacent The smallest angle is the smallest side. tan X = 7 24 X = tan X = Opposite 10 Marks

10 2016 JCHL Paper 2 Question 12 (b) (i) A different triangular-based prism has the base shown in the diagram on the right. Use trigonometry to find the length of the side marked x cm. Give your answer correct to two decimal places. Hypotenuse x Adjacent cos 70 = 3.5 x x = 3.5 cos 70 x = cm Marks

11 2015 JCHL Paper 2 Question 8 (b) ST = 10 and RS = 30. Using this information, and trigonometry, find the size of X. Give your answer in degrees, correct to one decimal place. X 30 Hypotenuse Opposite 10 sin X = X = sin 30 X = Marks

12 2015 JCHL Paper 2 Question 13 (a) Miriam is trying to find the volume of the water tank shown in the photograph on the right. She takes some measurements and draws a diagram. Part of her diagram is shown below. Using the diagram, find the value of x. Give your answer in metres, correct to two decimal places. Opposite x Adjacent tan 30 = x 20 x = 20 tan 30 x = m Source: Altered. 5 Marks

13 2015 JCHL Paper 2 Question 13 (b) The angle of elevation to the bottom of the water tank is 30, as shown in the diagram. The angle of elevation to the top of the water tank is 38. Find the distance marked h on the photograph. Give your answer correct to one decimal place. x Opposite x Adjacent h = h = 4.08 m tan 38 = x 20 x = 20 tan 38 x = m 10 Marks

14 2014 JCHL Paper 2 Question 6 (i) Construct a right angled triangle ABC, where: AB = 6 cm ABC = 90 AC = 10 cm. 10 Marks

15 2014 JCHL Paper 2 Question 6 (ii) On your diagram, measure the angle CAB. Give your answer correct to the nearest degree. CAB = 53 (iii) Let X be the whole number you wrote as your answer to (ii). Use a calculator to find cos X. Give your answer correct to 3 decimal places. X = 53 X = 53 cos 53 = Marks

16 2014 JCHL Paper 2 Question 6 (iv) Jacinta says that cos( CAB) is exactly 0.6, because cos( CAB) = Explain why your answer in (iii) is not the same as Jacinta s. cos( CAB) = 6 10 cos( CAB) = 0.6 CAB = cos CAB = So if X is a whole number then cos X can never be exactly 0.6. Hypotenuse CAB Adjacent 5 Marks

17 2014 JCHL Paper 2 Question 7 (v) Madison draws the scale diagram of the triangle OAB shown on the right. She marks in the angle X. Recall that [AB] is a metal bar, which is part of the frame of the swing. Write down the value of tan X, and hence find the size of the angle X. Give the size of the angle X correct to two decimal places. Opposite tan X = 5 4 X = tan X = Adjacent 10 Marks

18 2014 JCHL Paper 2 Question 7 (vi) In order to increase the height of the swing, it is decided to increase X by 20%. The distance AB will be kept the same. Find the new height of the swing. Give your answer in metres, correct to one decimal place. Adjacent Opposite X = X = X = sin X = sin = h sin = h h = m 10 Marks

19 2014 Sample JCHL Paper 2 Question 15 During a trigonometry lesson a group of students wrote down some statements about what they expected to happen when they looked at the values of trigonometric functions of some angles. Here are some of the things they wrote down. (i) The value from any of these trigonometric functions will always be less than 1. (ii) If the size of the angle is doubled then the value from the trigonometric functions will not double. (iii) The value from all of the trigonometric functions will increase if the size of the angle is increased. (iv) I do not need to use a calculator to find sin 60. I can do it by drawing an equilateral triangle. The answer will be in surd form. They then found the sin, cos and tan of some angles, correct to three decimal places, to test their ideas. (a) Do you think that (i) is correct? Give an example to justify your answer. No it IS possible for the side to be greater than the. e.g. tan 50 = 1.19 (b) Do you think that (ii) is correct? Give an example to justify your answer. (c) Do you think that (iii) is correct? Give an example to justify your answer. Yes the value of the ratio does NOT double when the angle does. e.g. tan 20 = 0.36 tan 40 = 0.83 No - not all of the ratios increase when the angle does. cos 20 = 0.94 cos 40 = 0.77

20 2014 Sample JCHL Paper 2 Question 15 During a trigonometry lesson a group of students wrote down some statements about what they expected to happen when they looked at the values of trigonometric functions of some angles. Here are some of the things they wrote down. (i) The value from any of these trigonometric functions will always be less than 1. (ii) If the size of the angle is doubled then the value from the trigonometric functions will not double. (iii) The value from all of the trigonometric functions will increase if the size of the angle is increased. (iv) I do not need to use a calculator to find sin 60. I can do it by drawing an equilateral triangle. The answer will be in surd form. They then found the sin, cos and tan of some angles, correct to three decimal places, to test their ideas. (d) Show how an equilateral triangle of side 2 cm can be used to find sin 60 in surd form. 2 2 x Pythagoras c 2 = a 2 + b = x = x x 2 = 4 1 x 2 = 3 x = 3 sin 60 = sin 60 =

21 2014 Sample JCHL Paper 2 Question 16 The Leaning Tower of Pisa is m tall and leans 3.9 m from the perpendicular, as shown below. The tower of the Suurhusen Church in north-western Germany is m tall and leans 2.47 m from the perpendicular. By providing diagrams and suitable calculations and explanations, decide which tower should enter the Guinness Book of Records as the Most Tilted Tower in the World Opposite Opposite x x 3.9 Adjacent 2.47 Adjacent tan x = x = tan 1 x = 86 4 lean 3.9 tan x = x = tan x = lean MOST TILTED TOWER

22 2014 Sample JCHL Paper 2 Question 17 (i) In the right-angled triangle shown in the diagram, one of the acute angles is four times as large as the other acute angle. Find the measures of the two acute angles in the triangle. x 4x The angles of a triangle sum to x + 4x = 180 5x = 90 x = 18 4x = 4 18 = 72

23 2014 Sample JCHL Paper 2 Question 17 (ii) The triangle in part (i) is placed on a co-ordinate diagram. The base is parallel to the x-axis. Find the slope of the line l that contains the of the triangle. Give your answer correct to three decimal places. 18 l base run rise slope = rise run slope = tan 18 = tan 18 =

24 2013 JCHL Paper 2 Question 10 (a) In the triangle ABC, AB = 2 and BC = 1. Find AC, giving your answer in surd form. (b) Write cos BAC and hence find BAC. A BAC 2 AC B C 1 cos BAC = 3 2 BAC = cos BAC = 30 Pythagoras c 2 = a 2 + b = x 2 4 = 1 + x = x 2 3 = x 2 3 = AC 5 Marks 5 Marks

25 2013 JCHL Paper 2 Question 10 (c) Sketch a right angled isosceles triangle in which the equal sides are 1 unit each and use it to write cos 45 in surd form. 1 1 x 45 Pythagoras c 2 = a 2 + b 2 x 2 = x 2 = x 2 = 2 x = 2 cos 45 = Marks (d) Show that cos 75 cos 45 + cos 30. cos 75 cos 45 + cos Marks

26 2013 JCHL Paper 2 Question 13 A tree 32 m high casts a shadow 63 m long. Calculate θ, the angle of elevation of the sun. Give your answer in degrees and minutes (correct to the nearest minute). Opposite 32 m tan θ = tan θ = θ = tan θ = θ = Adjacent 10 Marks

27 2012 JCHL Paper 2 Question 12 A homeowner wishes to replace the three identical steps leading to her front door with a ramp. Each step is 10 cm high and 35 cm long. Find the length of the ramp. Give your answer correct to one decimal place. 10 cm Height = 3 10 = 30 cm Width = 3 35 = 105 cm 30 x Pythagoras c 2 = a 2 + b cm x 2 = x 2 = x 2 = x = cm 10 Marks

28 2012 JCHL Paper 2 Question 13 (a) Two vertical poles A and B, each of height h, are standing on sides of a level road. They are 24 m apart. The point P, on the road directly between the two poles, is a distance x from pole A. The angle of elevation from P to the top of pole A is 60. Write h in terms of x. Adjacent Opposite tan 60 = h x h = x tan 60 h = x 3 h = 3x 5 Marks

29 2012 JCHL Paper 2 Question 13 (b) From P the angle of elevation to the top of pole B is 30. Find h, the height of the two poles. Opposite tan 30 = h 24 x h = 24 x tan 30 3 h = 24 x 3 h = 24 x x Let the value of h from (a) and (b) Adjacent equal and solve for x. 3x = 24 x x = 24 x 3 3x = 24 x 3x + x = 24 4x = 24 x = 6 24 x 3 h = 3 h = h = h = m 2 Marks

30 2011 JCHL Paper 2 Question 15 (a) A group of students wish to calculate the height of the Millennium Spire in Dublin. The spire stands on flat level ground. Maria, who is 1.72 m tall, looks up at the top of the spire using a clinometer and records an angle of elevation of 60. Her feet are 70 m from the base of the spire. Ultan measures the circumference of the base of the spire as 7.07 m. Explain how Ultan s measurement will be used in the calculation of the height of the Spire. The circumference can be used to calculate the radius, which will give the full distance that Maria is from the centre of the base of the spire. Circumference of a Circle = 2πr πr = 7.07 r = π r = r 1.13 m r 5 Marks for both (a) AND (b)!

31 2011 JCHL Paper 2 Question 15 (b) Draw a suitable diagram and calculate the height of the spire, to the nearest metre, using measurements obtained by the students. tan 60 = h h = tan 60 = = m h h 1.72 m Maria m 1.13