TRIGONOMETRY - Maths Points
Transcript of TRIGONOMETRY - Maths Points
TRIGONOMETRYSINE AND COSINE RULES
& AREA OF TRIANGLE
Leaving Cert Revision
Find the distance ๐ฅ in the diagram below (not to scale).Give your answer correct to 2 decimal places.
2017 LCOL Paper 2 โ Question 6 (a)
First fill in the missing angle in the triangle. ๐๐๐ โ ๐๐ + ๐๐ = ๐๐
52ยฐ
Sine Rule๐
sin ๐ด=
๐
sin ๐ต
๐
sin ๐ด=
๐
sin ๐ต๐ฅ
sin 52=
10
sin 63
๐ฅ sin 63 = 10 sin 52
๐ฅ =10 sin 52
sin 63
๐ฅ = 8.84 cm
10 Marks
Find the distance ๐ฆ in the diagram below (not to scale).Give your answer correct to 2 decimal places.
2017 LCOL Paper 2 โ Question 6 (b)
Cosine Rule
๐2 = ๐2 + ๐2 โ 2๐๐ cos ๐ด
๐2 = ๐2 + ๐2 โ 2๐๐ cos ๐ด
๐ฆ2 = 10.22 + 8.52 โ 2 10.2 8.5 cos 53.8 ยฐ
๐ฆ2 = 73.88
๐ฆ = 73.88
๐ฆ = 8.6 cm
25 Marks
Find the area of the given triangle.
2016 LCOL Paper 2 โ Question 2 (a)
=1
28 12 sin 30
= 24 cm2
Area of a Triangle
=1
2๐๐ sin ๐ถ
5 Marks
7
3
5๐2 = ๐2 + ๐2 โ 2๐๐ cos ๐ด72 = 32 + 52 โ 2 3 5 cos ๐49 = 9 + 25 โ 30 cos ๐30 cos ๐ = 9 + 25 โ 49
cos ๐ = โ15
30
cos ๐ = โ1
2๐ = 120ยฐ
๐ยฐ
Cosine Rule
๐2 = ๐2 + ๐2 โ 2๐๐ cos ๐ด
.
A triangle has sides of length 3 cm, 5 cm, and 7 cm.Find the size of the largest angle in the triangle.
2016 LCOL Paper 2 โ Question 2 (b)
20 Marks
Joe wants to draw a diagram of his farm. He uses axes and co-ordinates to plot his farmhouse at the point ๐น on the diagram below.
Write down the co-ordinates of the point F.
2016 LCOL Paper 2 โ Question 9 (a) (i)
4,1
4,6๐ต
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 ๐ต.
(ii)
๐น = 4,1
5 units
Combination of Co-ordinateGeometry and Trigonometry
5 Marks
5 Marks
Joe's quad bike is marked with the point ๐ on the diagram.Find the distance from the barn (๐ต) to the quad (๐).Give your answer correct to 2 decimal places.
2016 LCOL Paper 2 โ Question 9 (b)
โ2,7
4,1
4,6๐ต
๐๐ต = 4 โ โ22
+ 6 โ 7 2
๐๐ต = 6 2 + โ1 2
๐๐ต = 36 + 1
๐๐ต = 37๐๐ต = 6.08 units
Distance
= ๐ฅ2 โ ๐ฅ12 + ๐ฆ2 โ ๐ฆ1
2๐ โ2,7๐ต 4,6
5 Marks
Joe's tractor is at the point ๐, where ๐น๐ต๐๐ is a parallelogram.Plot ๐ on the diagram and write the co-ordinates of ๐ in the space below.
2016 LCOL Paper 2 โ Question 9 (c)
โ2,7
4,1
4,6๐ต
๐
๐ต๐4,6 โ โ2,7
We can find the co-ordinates of ๐ป by finding the image
of ๐ญ under the translation ๐ฉ๐ธ.
โ 6, โ 1
4,1 โ โ2,2
5 Marks
Joe's tractor is at the point ๐, where ๐น๐ต๐๐ is a parallelogram.Plot ๐ on the diagram and write the co-ordinates of ๐ in the space below.
2016 LCOL Paper 2 โ Question 9 (d)
โ2,7
4,1
4,6๐ต
๐
Area of a ParallelogramA = base ร perpendicular height
Base= 5
Height = 6
Area = 5 ร 6= 30 units2
5 Marks
Given that |โ ๐๐น๐ต| = 45ยฐ, use trigonometric methods to find |โ ๐ต๐๐น|.Give your answer in degrees correct to one decimal place.
2016 LCOL Paper 2 โ Question 9 (e)
๐ต
45ยฐ
6.08
5
๐ยฐ๐
sin ๐ด=
๐
sin ๐ต
6.08
sin 45ยฐ=
5
sin ๐
sin ๐ =5 sin 45ยฐ
6.08
๐ = sinโ15 sin 45ยฐ
6.08
๐ = 35.6ยฐ
Sine Rule๐
sin ๐ด=
๐
sin ๐ต
20 Marks
The diagram shows the triangles ๐ต๐ถ๐ท and ๐ด๐ต๐ท, with some measurements given.
Find |๐ต๐ถ|, correct to two decimal places.
2015 LCOL Paper 2 โ Question 5 (a) (i)
16
sin 110=
๐ต๐ถ
sin 42
๐ต๐ถ =16 sin 42
sin 110
๐ต๐ถ = 11.39
11.39
Sine Rule๐
sin ๐ด=
๐
sin ๐ต
15 Marks
Find the area of the triangle ๐ต๐ถ๐ท, correct to two decimal places.
2015 LCOL Paper 2 โ Question 5 (a) (ii)
180 โ 42 + 110= 28ยฐ
11.39
28ยฐ
Area of a Triangle
=1
2๐๐ sin ๐ถ
=1
216 11.39 sin 28ยฐ
= 42.78 m2
First fill in the missing angle in the triangle, ๐ซ๐ซ๐ช๐ฉ.
5 Marks
Find |๐ด๐ต|, correct to two decimal places.
2015 LCOL Paper 2 โ Question 5 (b)
180 โ 63 + 42= 75
75
16.53
First fill in the missing angle in the triangle, ๐ซ๐ซ๐จ๐ฉ.
Cosine Rule
๐2 = ๐2 + ๐2 โ 2๐๐ cos ๐ด
๐ด๐ต 2 = 102 + 162 โ 2 10 16 cos 75๐ด๐ต 2 = 273.18๐ด๐ต = 16.53 m
5 Marks
20 1822
25
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:๐ด๐ต = 30 cm๐ด๐ท = ๐ถ๐ต = 5 cm๐ถ๐น = 22 cm๐ธ๐น = 4 cm.
2014 LCOL Sample Paper 2 โ Question 8
25 2018
22
Two diagrams are given below โ one showing triangle ๐ถ๐ด๐น and the other showing triangle ๐ถ๐ท๐ธ. Use the measurements given above to record on the two diagrams below the lengths of two of the sides in each triangle.
2014 LCOL Sample Paper 2 โ Question 8 (a)
Taking ฮฑ = 60ยฐ, as shown, use the triangle ๐ถ๐ด๐น to find โ ๐ถ๐น๐ด , correct to one decimal place.
2014 LCOL Sample Paper 2 โ Question 8 (b)
25
25
sin ๐ฅ=
22
sin 60
sin ๐ฅ =25 sin 60
22๐ฅ = 79.78ยฐ
180 โ 79.78 โ 60 = 40.22ยฐ
Hence find โ ๐ด๐ถ๐น , correct to one decimal place.
(c)
Sine Rule๐
sin ๐ด=
๐
sin ๐ต
20 1822
40.22ยฐ
60ยฐ ๐ฅยฐ
Use triangle ๐ถ๐ท๐ธ to find ๐ท๐ธ , the length of the strap, correct to one decimal place.
2014 LCOL Sample Paper 2 โ Question 8 (d)
60ยฐ
25
79.78ยฐ
๐ท๐ธ 2 = 202 + 182 โ 2 20 18 cos 40.22ยฐ๐ท๐ธ 2 = 174.23๐ท๐ธ = 13.2
Cosine Rule
๐2 = ๐2 + ๐2 โ 2๐๐ cos ๐ด
20 1822
40.22ยฐ
A triangle in which the three sides have different lengths.
The planned supports for the roof of a building form scalene triangles of different sizes.Explain what is meant by a scalene triangle.
2012 Paper 2 โ Question 7 (a)
5 Marks
The triangle ๐ธ๐น๐บ is the image of the triangle ๐ถ๐ท๐ธ under an enlargement and the triangle ๐ถ๐ท๐ธ is the image of the triangle ๐ด๐ต๐ถ under the same enlargement.The proposed dimensions for the structure are ๐ด๐ต = 7.2 m, ๐ต๐ถ = 8 m, |๐ถ๐ท| = 9 m and |โ ๐ท๐ถ๐ต| = 60ยฐ .
Find the length of [๐น๐บ].
2012 Paper 2 โ Question 7 (b)
๐ =9
7.2๐ = 1.25
๐น๐บ = 8 ร 1.25 ร 1.25= 12.5 m
๐๐๐๐ฅ๐ ๐ ๐๐๐ญ๐จ๐ซ
๐ =Image Length
Object Length
15 Marks
Find the length of [๐ต๐ท], correct to three decimal places.
2012 Paper 2 โ Question 7 (c)
๐ต๐ท 2 = 82 + 92 โ 2 8 9 cos 60ยฐ๐ต๐ท 2 = 73
๐ต๐ท = 73๐ต๐ท = 8.544 m
Cosine Rule
๐2 = ๐2 + ๐2 โ 2๐๐ cos ๐ด
15 Marks
The centre of the enlargement is ๐. Find the distance from ๐ to the point ๐ต.
2012 Paper 2 โ Question 7 (d)
๐๐ท
๐๐ต= 1.25
๐ฅ + 8.544
๐ฅ= 1.25
๐ฅ + 8.544 = 1.25๐ฅ0.25๐ฅ = 8.544๐ฅ = 34.176 m
๐
๐ฅ
8.544
Scale Factor = 1.25
L๐๐ญ ๐ be the section from O to D, ๐ถ๐ซ
5 Marks
A condition of the planning is that the height of the point ๐บ above the horizontal line ๐ต๐น cannot exceed 11.6 m. Does the plan meet this condition? Justify your answer by calculation.
2012 Paper 2 โ Question 7 (e)
โ12.5
๐ผ๐ผ
9
sin ๐ผ=
8.544
sin 60
sin ๐ผ =9 sin 60
8.544
โ
sin ๐ผ=
12.5
sin 90
sin ๐ผ =โ sin 90
12.5
โ sin 90
12.5=
9 sin 60
8.544โ 1
12.5=
9 sin 60
8.5448.544โ = 12.5 9 sin 60
โ =12.5 9 sin 60
8.544โ = 11.4
11.4 < 11.6
Yes, the plan meets the condition. Sine Rule
๐
sin ๐ด=
๐
sin ๐ต
Sine Rule๐
sin ๐ด=
๐
sin ๐ต
10 Marks