TRIGONOMETRY - Maths Points
Transcript of TRIGONOMETRY - Maths Points
TRIGONOMETRYSINE AND COSINE RULES
& AREA OF TRIANGLE
Leaving Cert Revision
Find the distance π₯ in the diagram below (not to scale).Give your answer correct to 2 decimal places.
2017 LCOL Paper 2 β Question 6 (a)
First fill in the missing angle in the triangle. πππ β ππ + ππ = ππ
52Β°
Sine Ruleπ
sin π΄=
π
sin π΅
π
sin π΄=
π
sin π΅π₯
sin 52=
10
sin 63
π₯ sin 63 = 10 sin 52
π₯ =10 sin 52
sin 63
π₯ = 8.84 cm
10 Marks
Find the distance π¦ in the diagram below (not to scale).Give your answer correct to 2 decimal places.
2017 LCOL Paper 2 β Question 6 (b)
Cosine Rule
π2 = π2 + π2 β 2ππ cos π΄
π2 = π2 + π2 β 2ππ cos π΄
π¦2 = 10.22 + 8.52 β 2 10.2 8.5 cos 53.8 Β°
π¦2 = 73.88
π¦ = 73.88
π¦ = 8.6 cm
25 Marks
Find the area of the given triangle.
2016 LCOL Paper 2 β Question 2 (a)
=1
28 12 sin 30
= 24 cm2
Area of a Triangle
=1
2ππ sin πΆ
5 Marks
7
3
5π2 = π2 + π2 β 2ππ cos π΄72 = 32 + 52 β 2 3 5 cos π49 = 9 + 25 β 30 cos π30 cos π = 9 + 25 β 49
cos π = β15
30
cos π = β1
2π = 120Β°
πΒ°
Cosine Rule
π2 = π2 + π2 β 2ππ cos π΄
.
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 β 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 πΉ on the diagram below.
Write down the co-ordinates of the point F.
2016 LCOL Paper 2 β Question 9 (a) (i)
4,1
4,6π΅
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 π΅.
(ii)
πΉ = 4,1
5 units
Combination of Co-ordinateGeometry and Trigonometry
5 Marks
5 Marks
Joe's quad bike is marked with the point π on the diagram.Find the distance from the barn (π΅) to the quad (π).Give your answer correct to 2 decimal places.
2016 LCOL Paper 2 β Question 9 (b)
β2,7
4,1
4,6π΅
ππ΅ = 4 β β22
+ 6 β 7 2
ππ΅ = 6 2 + β1 2
ππ΅ = 36 + 1
ππ΅ = 37ππ΅ = 6.08 units
Distance
= π₯2 β π₯12 + π¦2 β π¦1
2π β2,7π΅ 4,6
5 Marks
Joe's tractor is at the point π, where πΉπ΅ππ is a parallelogram.Plot π on the diagram and write the co-ordinates of π in the space below.
2016 LCOL Paper 2 β Question 9 (c)
β2,7
4,1
4,6π΅
π
π΅π4,6 β β2,7
We can find the co-ordinates of π» by finding the image
of π under the translation π©πΈ.
β 6, β 1
4,1 β β2,2
5 Marks
Joe's tractor is at the point π, where πΉπ΅ππ is a parallelogram.Plot π on the diagram and write the co-ordinates of π in the space below.
2016 LCOL Paper 2 β Question 9 (d)
β2,7
4,1
4,6π΅
π
Area of a ParallelogramA = base Γ perpendicular height
Base= 5
Height = 6
Area = 5 Γ 6= 30 units2
5 Marks
Given that |β ππΉπ΅| = 45Β°, use trigonometric methods to find |β π΅ππΉ|.Give your answer in degrees correct to one decimal place.
2016 LCOL Paper 2 β Question 9 (e)
π΅
45Β°
6.08
5
πΒ°π
sin π΄=
π
sin π΅
6.08
sin 45Β°=
5
sin π
sin π =5 sin 45Β°
6.08
π = sinβ15 sin 45Β°
6.08
π = 35.6Β°
Sine Ruleπ
sin π΄=
π
sin π΅
20 Marks
The diagram shows the triangles π΅πΆπ· and π΄π΅π·, with some measurements given.
Find |π΅πΆ|, correct to two decimal places.
2015 LCOL Paper 2 β Question 5 (a) (i)
16
sin 110=
π΅πΆ
sin 42
π΅πΆ =16 sin 42
sin 110
π΅πΆ = 11.39
11.39
Sine Ruleπ
sin π΄=
π
sin π΅
15 Marks
Find the area of the triangle π΅πΆπ·, correct to two decimal places.
2015 LCOL Paper 2 β Question 5 (a) (ii)
180 β 42 + 110= 28Β°
11.39
28Β°
Area of a Triangle
=1
2ππ sin πΆ
=1
216 11.39 sin 28Β°
= 42.78 m2
First fill in the missing angle in the triangle, π«π«πͺπ©.
5 Marks
Find |π΄π΅|, correct to two decimal places.
2015 LCOL Paper 2 β Question 5 (b)
180 β 63 + 42= 75
75
16.53
First fill in the missing angle in the triangle, π«π«π¨π©.
Cosine Rule
π2 = π2 + π2 β 2ππ cos π΄
π΄π΅ 2 = 102 + 162 β 2 10 16 cos 75π΄π΅ 2 = 273.18π΄π΅ = 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:π΄π΅ = 30 cmπ΄π· = πΆπ΅ = 5 cmπΆπΉ = 22 cmπΈπΉ = 4 cm.
2014 LCOL Sample Paper 2 β Question 8
25 2018
22
Two diagrams are given below β one showing triangle πΆπ΄πΉ and the other showing triangle πΆπ·πΈ. 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 β Question 8 (a)
Taking Ξ± = 60Β°, as shown, use the triangle πΆπ΄πΉ to find β πΆπΉπ΄ , correct to one decimal place.
2014 LCOL Sample Paper 2 β Question 8 (b)
25
25
sin π₯=
22
sin 60
sin π₯ =25 sin 60
22π₯ = 79.78Β°
180 β 79.78 β 60 = 40.22Β°
Hence find β π΄πΆπΉ , correct to one decimal place.
(c)
Sine Ruleπ
sin π΄=
π
sin π΅
20 1822
40.22Β°
60Β° π₯Β°
Use triangle πΆπ·πΈ to find π·πΈ , the length of the strap, correct to one decimal place.
2014 LCOL Sample Paper 2 β Question 8 (d)
60Β°
25
79.78Β°
π·πΈ 2 = 202 + 182 β 2 20 18 cos 40.22Β°π·πΈ 2 = 174.23π·πΈ = 13.2
Cosine Rule
π2 = π2 + π2 β 2ππ cos π΄
20 1822
40.22Β°
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 β Question 7 (a)
5 Marks
The triangle πΈπΉπΊ is the image of the triangle πΆπ·πΈ under an enlargement and the triangle πΆπ·πΈ is the image of the triangle π΄π΅πΆ under the same enlargement.The proposed dimensions for the structure are π΄π΅ = 7.2 m, π΅πΆ = 8 m, |πΆπ·| = 9 m and |β π·πΆπ΅| = 60Β° .
Find the length of [πΉπΊ].
2012 Paper 2 β Question 7 (b)
π =9
7.2π = 1.25
πΉπΊ = 8 Γ 1.25 Γ 1.25= 12.5 m
ππππ₯π π ππππ¨π«
π =Image Length
Object Length
15 Marks
Find the length of [π΅π·], correct to three decimal places.
2012 Paper 2 β Question 7 (c)
π΅π· 2 = 82 + 92 β 2 8 9 cos 60Β°π΅π· 2 = 73
π΅π· = 73π΅π· = 8.544 m
Cosine Rule
π2 = π2 + π2 β 2ππ cos π΄
15 Marks
The centre of the enlargement is π. Find the distance from π to the point π΅.
2012 Paper 2 β Question 7 (d)
ππ·
ππ΅= 1.25
π₯ + 8.544
π₯= 1.25
π₯ + 8.544 = 1.25π₯0.25π₯ = 8.544π₯ = 34.176 m
π
π₯
8.544
Scale Factor = 1.25
Lππ π be the section from O to D, πΆπ«
5 Marks
A condition of the planning is that the height of the point πΊ above the horizontal line π΅πΉ cannot exceed 11.6 m. Does the plan meet this condition? Justify your answer by calculation.
2012 Paper 2 β Question 7 (e)
β12.5
πΌπΌ
9
sin πΌ=
8.544
sin 60
sin πΌ =9 sin 60
8.544
β
sin πΌ=
12.5
sin 90
sin πΌ =β sin 90
12.5
β sin 90
12.5=
9 sin 60
8.544β 1
12.5=
9 sin 60
8.5448.544β = 12.5 9 sin 60
β =12.5 9 sin 60
8.544β = 11.4
11.4 < 11.6
Yes, the plan meets the condition. Sine Rule
π
sin π΄=
π
sin π΅
Sine Ruleπ
sin π΄=
π
sin π΅
10 Marks