13 Jun 2022 13 Jun 2022 Created by Mr. Lafferty Maths Dept. Created by Mr. Lafferty Maths Dept. Exact Values Angles greater than 90 o Trigonometry Trigonometry www.mathsrevision.com Useful Notation & Area of a triangl Using Area of Triangle Formula Cosine Rule Problems Sine Rule Problems Mixed Problems S5 Int2
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12 Apr 202312 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Exact Values
Angles greater than 90o
TrigonometryTrigonometryw
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Useful Notation & Area of a triangle
Using Area of Triangle Formula
Cosine Rule Problems
Sine Rule Problems
Mixed Problems
S5 Int2
12 Apr 202312 Apr 2023 Created by Mr Lafferty Maths DeptCreated by Mr Lafferty Maths Dept
Starter QuestionsStarter Questions
o
21. Factorise x - 36
2. A car depreciates at 20% each year.
How much is it worth af ter 4 years if it cost
£ 15 000 initially.
3. What sin30 as a f raction.
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S5 Int2
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1.1. Recognise basic triangles Recognise basic triangles and exact values for sin, and exact values for sin, cos and tan 30cos and tan 30oo, 45, 45oo, 60, 60oo . .
1. To build on basic trigonometry values.
2.2. Calculate exact values for Calculate exact values for problems.problems.
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S5 Int2
22
2
60º
60º
60º 1
60º
230º3
This triangle will provide exact values for
sin, cos and tan 30º and 60º
Exact ValuesExact Valuesw
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S5 Int2
Some special values of Sin, Cos and Tan are useful left as fractions, We call these exact values
x 0º 30º 45º 60º 90º
Sin xº
Cos xº
Tan xº
½
½
3
3
2
3
20
1
0
1
0
Exact ValuesExact Valuesw
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S5 Int2
1
3
Exact ValuesExact Valuesw
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S5 Int2
1 145º
45º
2
Consider the square with sides 1 unit
11
We are now in a position to calculate exact values for sin, cos and tan of 45o
x 0º 30º 45º 60º 90º
Sin xº
Cos xº
Tan xº
½
½
3
3
2
3
20
1
0
1
0
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S5 Int2
1
3
1 2
1 2
1
12 Apr 202312 Apr 2023 Created by Mr Lafferty Maths DeptCreated by Mr Lafferty Maths Dept
Now try Exercise 1Ch8 (page 94)
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S5 Int2
Exact ValuesExact Values
12 Apr 202312 Apr 2023 Created by Mr Lafferty Maths DeptCreated by Mr Lafferty Maths Dept
Starter QuestionsStarter Questions
1. Write down the Compound I nterest Formula
and identif y each term.
2. A house increases by 3% each year.
How much is it worth in 5 years if it cost
£ 40 000 initially.
3. What is the .oexact value of sin 45
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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. To explain how to use the Area formula for ANY triangle.
S5 Int2
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.
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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S5 Int2
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
knowing 2 sides and the
angle in between (SAS)
12 Apr 202312 Apr 2023 Created by Mr Lafferty Maths DeptCreated by Mr Lafferty Maths Dept
Area of ANY TriangleArea of ANY Trianglew
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S5 Int2
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 sin30
2oArea
210 25 0.5 125Area cm
12 Apr 202312 Apr 2023 Created by Mr Lafferty Maths DeptCreated by Mr Lafferty Maths Dept
Area of ANY TriangleArea of ANY Trianglew
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S5 Int2
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)
12 Apr 202312 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Now try Exercise 4Ch8 (page 100)
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Area of ANY TriangleArea of ANY Triangle
12 Apr 202312 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Starter QuestionsStarter Questionsw
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2
1. Multiply out the brackets and simplif y
5(y- 5) - 7(5- y)
2. Find the gradient and the y - intercept
3 f or the line with equation y = 5x -
4
3. Factorise x -100
S5 Int2
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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 problems involving lengths.lengths.
1. To show how to use the sine rule to solve REAL LIFE problems involving finding the length of a side of a triangle .
Sine RuleSine RuleS5 Int2
C
B
A12 Apr 202312 Apr 2023 Created by Mr Lafferty Maths DeptCreated by Mr Lafferty Maths Dept
Sine RuleSine Rulew
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S5 Int2
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
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
sin34
o
oa 10 0.656
11.740.559
a m
Example 1 : Find the length of a in this triangle.
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S4 Credit
A
B
C
sin sin sino
a b c
A B C
Calculating Sides Calculating Sides Using The Sine Using The Sine
RuleRule
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S4 Credit
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
What goes in the Box What goes in the Box ??
Find the unknown side in each of the triangles below:
(1) 12cm
72o
32o
a
(2)
93o
b47o
16mm
a = 6.7cm b =
21.8mm
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12 Apr 202312 Apr 2023 Created by Mr Lafferty Maths DeptCreated by Mr Lafferty Maths Dept
12 Apr 202312 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Now try Ex 6&7 Ch8 (page 103)
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Sine RuleSine Rule
12 Apr 202312 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Starter QuestionsStarter Questionsw
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1. Factorise 9x - 36
2. Find the gradient and the y - intercept
3 1 f or the line with equation y = - x +
4 5
3. Write down the two values of cos
1 that give you a value of
2
S5 Int2
12 Apr 202312 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.ww
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. To use our knowledge gained so far to solve various trigonometry problems.
Mixed problemsMixed problemsS5 Int2
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.
Find the bearing of Q from point P.
2 2 2530 670 5202 530 670
CosPx x
48.7oP
180 22948.7 oBearing
P
670 miles
W
530 miles
Not to Scale
Q
520 miles
Exam Type Questions
12 Apr 202312 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.