TRIGONOMETRY SINE AND COSINE RULES & AREA OF TRIANGLE. Leaving Cert Revision

Transcription

1 TRIGONOMETRY SINE AND COSINE RULES & AREA OF TRIANGLE Leaving Cert Revision

2 2017 LCOL Paper 2 Question 6 (a) Find the distance x in the diagram below (not to scale). Give your answer correct to 2 decimal places. First fill in the missing angle in the triangle = 52 Sine Rule a sin A = b sin B 52 a sin A = b sin B x sin 52 = 10 sin 63 x sin 63 = 10 sin sin 52 x = sin 63 x = 8.84 cm 10 Marks

3 2017 LCOL Paper 2 Question 6 (b) Find the distance y in the diagram below (not to scale). Give your answer correct to 2 decimal places. Cosine Rule a 2 = b 2 + c 2 2bc cos A a 2 = b 2 + c 2 2bc cos A y 2 = cos 53.8 y 2 = y = y = 8.6 cm 25 Marks

4 2016 LCOL Paper 2 Question 2 (a) Find the area of the given triangle. Area of a Triangle = 1 ab sin C 2 = sin 30 = 24 cm 2 5 Marks

5 2016 LCOL Paper 2 Question 2 (b). A triangle has sides of length 3 cm, 5 cm, and 7 cm. Find the size of the largest angle in the triangle. 5 X 3 7 Cosine Rule a 2 = b 2 + c 2 2bc cos A a 2 = b 2 + c 2 2bc cos A 7 2 = cos X 49 = cos X 30 cos X = cos X = cos X = 1 2 X = Marks

6 2016 LCOL Paper 2 Question 9 (a) (i) Combination of Co-ordinate Geometry and Trigonometry Joe wants to draw a diagram of his farm. He uses axes and co-ordinates to plot his farmhouse at the point F on the diagram below. Write down the co-ordinates of the point F. 5 Marks F = 4,1 (ii) A barn is 5 units directly North of the farmhouse. Plot the point representing the position of the barn on the diagram. Label this point B. 5 Marks B 4,6 5 units 4,1

7 2016 LCOL Paper 2 Question 9 (b) Joe's quad bike is marked with the point Q on the diagram. Find the distance from the barn (B) to the quad (Q). Give your answer correct to 2 decimal places. 2,7 B 4,6 Distance = x 2 x y 2 y 1 2 QB = QB = QB = QB = 37 QB = 6.08 units Q 2,7 B 4,6 4,1 5 Marks

8 2016 LCOL Paper 2 Question 9 (c) Joe's tractor is at the point T, where FBQT is a parallelogram. Plot T on the diagram and write the co-ordinates of T in the space below. 2,7 We can find the co-ordinates of T by finding the image of F under the translation BQ. B 4,6 BQ 4,6 2,7 6, 1 4,1 2,2 T 4,1 5 Marks

9 2016 LCOL Paper 2 Question 9 (d) Joe's tractor is at the point T, where FBQT is a parallelogram. Plot T on the diagram and write the co-ordinates of T in the space below. 2,7 Area of a Parallelogram A = base perpendicular height B 4,6 Area = 5 6 = 30 units 2 Base = 5 T Height = 6 4,1 5 Marks

10 2016 LCOL Paper 2 Question 9 (e) Given that QFB = 45, use trigonometric methods to find BQF. Give your answer in degrees correct to one decimal place. X 6.08 B 45 5 Sine Rule a sin A = b sin B a sin A = b sin B 6.08 sin 45 = 5 sin X 5 sin 45 sin X = sin 45 X = sin X = Marks

11 2015 LCOL Paper 2 Question 5 (a) (i) The diagram shows the triangles BCD and ABD, with some measurements given. Find BC, correct to two decimal places. Sine Rule a sin A = b sin B sin 110 = BC sin sin 42 BC = sin 110 BC = Marks

12 2015 LCOL Paper 2 Question 5 (a) (ii) Find the area of the triangle BCD, correct to two decimal places. First fill in the missing angle in the triangle, ΔDCB = Area of a Triangle = 1 ab sin C 2 28 = sin 28 = m 2 5 Marks

13 2015 LCOL Paper 2 Question 5 (b) Find AB, correct to two decimal places. 75 First fill in the missing angle in the triangle, ΔDAB = 75 Cosine Rule a 2 = b 2 + c 2 2bc cos A AB 2 = cos 75 AB 2 = AB = m 5 Marks

14 2014 LCOL Sample Paper 2 Question 8 A stand is being used to prop up a portable solar panel. It consists of a support that is hinged to the panel near the top, and an adjustable strap joining the panel to the support near the bottom. By adjusting the length of the strap, the angle between the panel and the ground can be changed. The dimensions are as follows: AB = 30 cm AD = CB = 5 cm CF = 22 cm EF = 4 cm

15 2014 LCOL Sample Paper 2 Question 8 (a) Two diagrams are given below one showing triangle CAF and the other showing triangle CDE. Use the measurements given above to record on the two diagrams below the lengths of two of the sides in each triangle

16 2014 LCOL Sample Paper 2 Question 8 (b) Taking α = 60, as shown, use the triangle CAF to find CFA, correct to one decimal place. Sine Rule a sin A = b sin B 25 sin x = 22 sin sin 60 sin x = 22 x = (c) Hence find ACF, correct to one decimal place. 60 x = 40.22

17 2014 LCOL Sample Paper 2 Question 8 (d) Use triangle CDE to find DE, the length of the strap, correct to one decimal place. Cosine Rule a 2 = b 2 + c 2 2bc cos A DE 2 = cos DE 2 = DE =

18 2012 Paper 2 Question 7 (a) The planned supports for the roof of a building form scalene triangles of different sizes. Explain what is meant by a scalene triangle. A triangle in which the three sides have different lengths. 5 Marks

19 2012 Paper 2 Question 7 (b) The triangle EFG is the image of the triangle CDE under an enlargement and the triangle CDE is the image of the triangle ABC under the same enlargement. The proposed dimensions for the structure are AB = 7.2 m, BC = 8 m, CD = 9 m and DCB = 60. Find the length of [FG]. Scale Factor Image Length k = Object Length k = k = 1.25 FG = = 12.5 m 15 Marks

20 2012 Paper 2 Question 7 (c) Find the length of [BD], correct to three decimal places. Cosine Rule a 2 = b 2 + c 2 2bc cos A BD 2 = cos 60 BD 2 = 73 BD = 73 BD = m 15 Marks

21 2012 Paper 2 Question 7 (d) The centre of the enlargement is O. Find the distance from O to the point B. O x Let x be the section from O to D, OD Scale Factor = 1.25 OD OB = 1.25 x = 1.25 x x = 1.25x 0.25x = x = m 5 Marks

22 2012 Paper 2 Question 7 (e) A condition of the planning is that the height of the point G above the horizontal line BF cannot exceed 11.6 m. Does the plan meet this condition? Justify your answer by calculation. α 12.5 α h h sin 90 9 sin 60 = h 1 9 sin 60 = h = sin sin 60 h = h = < 11.6 Sine Rule a sin A = b sin B 9 sin α = sin 60 9 sin 60 sin α = Sine Rule a sin A = b sin B h sin α = 12.5 sin 90 h sin 90 sin α = 12.5 Yes, the plan meets the condition. 10 Marks