21 Mar 2022 21 Mar 2022 Created by Mr. Lafferty Maths Dept. Created by Mr. Lafferty Maths Dept. Trigonometry Trigonometry www.mathsrevision.com Cosine Rule Finding a Length Sine Rule Finding a length Mixed Problems Nat 5 Sine Rule Finding an Angle Cosine Rule Finding an Angle Area of ANY Triangle Revision (S O H)(C A H)(T O A) Exam Type Questions
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19 Apr 202319 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
TrigonometryTrigonometryw
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Cosine Rule Finding a Length
Sine Rule Finding a length
Mixed Problems
Nat 5
Sine Rule Finding an Angle
Cosine Rule Finding an Angle
Area of ANY Triangle
Revision (SOH)(CAH)(TOA)
Exam Type Questions
Starter QuestionsStarter Questions
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1. Calculate 104 x 100
putting your answer in standard f orm.
2. I s this triangle right angled ?
I f yes, fi nd the size of angle x .
I f no fi nd the area of the triangle.
xo
6
8
10
Nat 5
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1.1. Know the formula for the Know the formula for the area of any triangle.area of any triangle.
1. We are learning how to apply the Area formula for ANY triangle.
Nat 5
Area of ANY TriangleArea of ANY Triangle
2.2. Use formula to find area of Use formula to find area of any triangle given two any triangle given two length and angle in length and angle in between.between.
19 Apr 202319 Apr 2023 Created by Mr Lafferty Maths DeptCreated by Mr Lafferty Maths Dept
Labelling TrianglesLabelling Trianglesw
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A
B
C
A
aB
b
Cc
Small letters a, b, c refer to distancesCapital letters A, B, C refer to angles
In Mathematics we have a convention for labelling triangles.
F
E
D
F
E
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Labelling TrianglesLabelling Trianglesw
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d
e
f
Have a go at labelling the following triangle.
General Formula forGeneral Formula forArea of ANY TriangleArea of ANY Triangle
Consider the triangle below:
Ao Bo
Co
ab
c
h
Area = ½ x base x height 1
2A c h
What does the sine of Ao equal
sin o hA
b
Change the subject to h. h = b
sinAoSubstitute into the area formula
1sin
2oA c b A
1sin
2oA bc A
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Area of ANY TriangleArea of ANY Trianglew
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A
B
C
A
aB
b
Cc
The area of ANY triangle can be found by the following formula.
sin1
Area = ab C2
sin1
Area = ac B2
sin1
Area = bc A2
Another version
Another version
Key feature
To find the areayou need to know
2 sides and the angle in between
(SAS)
Demo
19 Apr 202319 Apr 2023 Created by Mr Lafferty Maths DeptCreated by Mr Lafferty Maths Dept
Area of ANY TriangleArea of ANY Trianglew
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A
B
C
A
20cmB
25cm
Cc
Example : Find the area of the triangle.
sinC1
Area = ab2
The version we use is
30o
120 25 sin 30
2oArea
210 25 0.5 125Area cm
19 Apr 202319 Apr 2023 Created by Mr Lafferty Maths DeptCreated by Mr Lafferty Maths Dept
Area of ANY TriangleArea of ANY Trianglew
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D
E
F
10cm
8cm
Example : Find the area of the triangle.
sin1
Area= df E2
The version we use is
60o
18 10 sin 60
2oArea
240 0.866 34.64Area cm
What Goes In The Box What Goes In The Box ??
Calculate the areas of the triangles below:
(1)
23o
15cm
12.6cm
(2)
71o
5.7m
6.2m
A = 36.9cm2
A = 16.7m2
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Key feature
Remember (SAS)
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TrigonomeTrigonometrytryNat 5
Now try N5 TJEx 8.2
Ch8 (page 73)
19 Apr 202319 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Starter QuestionsStarter Questionsw
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1. Multiply out the brackets and simplif y
5(y- 5) - 7(5- y)
2. True or f alse the gradient of the line is 5
3 y = 5x -
4
3. Factorise x -100
Nat 5
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1.1. Know how to use the sine Know how to use the sine rule to solve REAL LIFE rule to solve REAL LIFE problems involving lengths problems involving lengths showing ALL appropriate showing ALL appropriate working.working.
1. We are learning how to use the sine rule to solve REAL LIFE problems involving finding the length of a side of a triangle .
Sine RuleSine RuleNat 5
C
B
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Sine RuleSine Rulew
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a
b
c
The Sine Rule can be used with ANY triangle as long as we have been given enough information.
Works for any Triangle
a b c= =
SinA SinB SinC
Demo
Deriving the rule
B
C
A
b
c
a
Consider a general triangle ABC.
The Sine Rule
Draw CP perpendicular to BA
P
CPSinB CP aSinB
a
CP
also SinA CP bSinAb
aSinB bSinA
aSinBb
SinA
a bSinA SinB
This can be extended to
a b cSinA SinB SinC
or equivalentlySinA SinB SinCa b c
Calculating Sides Calculating Sides Using The Sine RuleUsing The Sine Rule
10m
34o
41o
a
Match up corresponding sides and angles:
sin 41oa
10
sin 34o
Rearrange and solve for a. 10sin 41
sin 34
o
oa 10 0.656
11.740.559
a m
Example 1 : Find the length of a in this triangle.
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A
B
C
sin sin sino
a b c
A B C
Demo
Calculating Sides Calculating Sides Using The Sine Using The Sine
RuleRule
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10m133o
37o
d
sin133od
10
sin 37o
10sin133
sin 37
o
od
10 0.731
0.602d
=
12.14m
Match up corresponding sides and angles:
Rearrange and solve for d.
Example 2 : Find the length of d in this triangle.
C
D
E
sin sin sino
c d e
C D E
Demo
What goes in the Box What goes in the Box ??
Find the unknown side in each of the triangles below:
(1)12cm
72o
32oa
(2)
93o
b47o
16mm
A = 6.7cm
B = 21.8mm
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TrigonomeTrigonometrytryNat 5
Now try N5 TJEx 8.3
Ch8 (page 76)
19 Apr 202319 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Starter QuestionsStarter Questionsw
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2. Find the gradient and the y - intercept
3 1 f or the line with equation y = - x +
4 5
3. Solve the equation tanx - 1 = 0
Nat 5
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1.1. Be able to recognise the Be able to recognise the correct trigonometric correct trigonometric formula to use to solve a formula to use to solve a problem involving problem involving triangles.triangles.
1. We are learning to use our knowledge gained so far to solve various trigonometry problems.
Mixed problemsMixed problemsNat 5
SOH CAH TOA
25o
15 mAD
The angle of elevation of the top of a building
measured from point A is 25o. At point D which is
15m closer to the building, the angle of elevation is
35o Calculate the height of the building.
T
B
Angle TDA =
145o
Angle DTA =
10o
o o
1525 10
TDSin Sin
o15 2536.5
10Sin
TD mSin
35o
36.5
o3536.5TB
Sin
o36.5 25 0. 93TB Sin m
180 – 35 = 145o
180 – 170 = 10o
sin sin sin
t d a
T D A
Exam Type Questions
A fishing boat leaves a harbour (H) and travels due East for 40 miles to a marker buoy (B). At B the boat turns left and sails for 24 miles to a lighthouse (L). It then returns to harbour, a distance of 57 miles.
(a) Make a sketch of the journey.
(b) Find the bearing of the lighthouse from the harbour. (nearest degree)
H40 miles
24 miles
B
L
57 miles
A
2 2 257 40 242 57 40
CosAx x
A 20.4o
90 0 020.4 7 oBearing
Exam Type Questions
A
The angle of elevation of the top of a column measured from point A, is 20o. The angle of elevation of the top of the statue is 25o. Find the height of the statue when the measurements are taken 50 m from its base
50 m
Angle BCA =
70o
Angle ACT = Angle ATC =
110o
65o
o 5020Cos
AC o
5020
ACCos
53.21 m
o o
53.215 65
TCSin Sin
o
53.21 5 (1 )
655.1
SinTC m dp
Sin
B
T
C
180 – 110 = 70o 180 – 70 = 110o 180 – 115 = 65o
20o
25o
5o
SOH CAH TOA
53.21 (2 )m dp
Exam Type Questions
sin sin sin
t d a
T D A
2 2 2
2b c a
CosAbc
An AWACS aircraft takes off from RAF Waddington (W) on a navigation exercise. It flies 530 miles North to a point (P) as shown, It then turns left and flies to a point (Q), 670 miles away. Finally it flies back to base, a distance of 520 miles.