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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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o
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
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2.2. Use Use SOHCAHTOA to finding inding an angle or length given an angle or length given a right-angled triangle.a right-angled triangle.
1. We are revising SOHCAHTOA process.
Angles & Angles & Triangles Triangles
1.1. Know the tree ratios for Know the tree ratios for SOHCAHTOA.
Nat 5
The Three RatiosThe Three Ratios
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Cosine
Sine
Tangent
Sine
Sine
Tangent
Cosine
Cosine
Sine
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opposite
opposite opposite
adjacent
adjacent
adjacent
hypotenuse
hypotenuse
hypotenuse
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TrigonomeTrigonometrytry
Sin x° =Opp
HypCos x° =
Adj
HypTan x° =
Opp
Adj
CAH TOASOH
Nat 5
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TrigonomeTrigonometrytry
SOH CAH TOA
Copy this!
1. Write down
Process
Identify what you want to find
what you know3.
2.
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TrigonomeTrigonometrytryNat 5
Now try N5 TJEx8.1 Q3 onwards
Ch8 (page 71)
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2( 3) (4 ) 3 2x x x 1. True or f alse
2. Find the equaton of the line passing
through the points ( 3,2) and (10, 9) .
3. Solve the equation sin x - 0.5 = 0
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.
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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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Nat 5
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
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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
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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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2
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
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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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Nat 5
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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19 Apr 202319 Apr 2023 Created by Mr Lafferty Maths DeptCreated by Mr Lafferty Maths Dept
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
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1.1. Know how to use the sine Know how to use the sine rule to solve problems rule to solve problems involving angles.involving angles.
1. We are learning how to use the sine rule to solve problems involving finding an angle of a triangle .
Sine RuleSine RuleNat 5
Calculating Angles Calculating Angles
Using The Sine Using The Sine RuleRule
Example 1 :
Find the angle Ao
A
45m
23o
38m
Match up corresponding sides and angles:
45
sin oA 38
sin 23o
Rearrange and solve for sin Ao
45sin 23sin
38
ooA = 0.463 Use sin-1 0.463 to find Ao
1sin 0.463 27.6o oA ww
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sin sin sin
a b c
A B C
B
C
Demo
Calculating Angles Calculating Angles
Using The Sine Using The Sine RuleRule
143o
75m
38m
X
38
sin oX
75
sin143o
38sin143sin
75
ooX = 0.305
1sin 0.305 17.8o oX
Example 2 :
Find the angle Xo
Match up corresponding sides and angles:
Rearrange and solve for sin Xo
Use sin-1 0.305 to find Xo
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Y
Z
sin sin sin
x y z
X Y Z
Demo
What Goes In The Box What Goes In The Box ??
Calculate the unknown angle in the following:
(1)
14.5m
8.9m
Ao
100o(2)
14.7cm
Bo
14o
12.9cm
Ao = 37.2o
Bo = 16ow
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2
1. Find the gradient of the line that passes
through the points ( 1,1) and (9,9).
2. Find the gradient and the y - intercept
f or the line with equation y = 1 - x
3. Factorise x - 64
Nat 5
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1.1. Know when to use the Know when to use the cosine rule to solve cosine rule to solve problems.problems.
1. We are learning when to use the cosine rule to solve problems involving finding the length of a side of a triangle .
Cosine RuleCosine RuleNat 5
2. 2. Solve problems that Solve problems that involve finding the length involve finding the length of a side.of a side.
C
B
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Cosine RuleCosine Rulew
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Nat 5
a
b
c
The Cosine Rule can be used with ANY triangle as long as we have been given enough information.
Works for any Triangle
cos2 2 2a =b +c - 2bc A
Deriving the rule
A
B
C
a
b
c
Consider a general triangle ABC. We require a in terms of b, c and A.
Draw BP perpendicular to AC
b
Px b - x
BP2 = a2 – (b – x)2
Also: BP2 = c2 – x2
a2 – (b – x)2 = c2 – x2
a2 – (b2 – 2bx + x2) = c2 – x2
a2 – b2 + 2bx – x2 = c2 – x2
a2 = b2 + c2 – 2bx*
a2 = b2 + c2 – 2bcCosA*Since Cos A = x/c x = cCosA
When A = 90o, CosA = 0 and reduces to a2 = b2 + c2
1
When A > 90o, CosA is negative, a2 > b2 + c2 2
When A < 90o, CosA is positive, a2 > b2 + c2 3
The Cosine Rule
The Cosine Rule generalises Pythagoras’ Theorem and takes care of the 3 possible cases for Angle A.
a2 > b2 + c2
a2 < b2 + c2
a2 = b2 + c2
A
A
A
1
2
3
Pythagoras + a bitPythagoras - a bit
Pythagoras
a2 = b2 + c2 – 2bcCosA
Applying the same method as earlier to the other sides produce similar formulae
for b and c. namely:b2 = a2 + c2 – 2acCosB
c2 = a2 + b2 – 2abCosC
A
B
C
a
b
c
The Cosine Rule
The Cosine rule can be used to find:
1. An unknown side when two sides of the triangle and the included angle are given (SAS).
2. An unknown angle when 3 sides are given (SSS).
Finding an unknown side.
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How to determine when to use the Cosine Rule.
Works for any Triangle
1. Do you know ALL the lengths.
2. Do you know 2 sides and the angle in between.
SASOR
If YES to any of the questions then Cosine Rule
Otherwise use the Sine Rule
Two questions
Using The Cosine Using The Cosine RuleRule
Example 1 : Find the unknown side in the triangle below: L5m
12m
43o
Identify sides a,b,c and angle Ao
a =
L b =
5 c =
12 Ao = 43o
Write down the Cosine Rule.
Substitute values to find a2.a2 =
52 + 122 - 2 x 5 x 12 cos 43o
a2 =
25 + 144
- (120 x
0.731 )
a2 =
81.28 Square root to find “a”.
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Demo
Example 2 :
Find the length of side M.
137o17.5 m
12.2 m
MIdentify the sides and angle.
a = M
b = 12.2 C = 17.5 Ao = 137o
Write down Cosine Rule
a2 = 12.22 + 17.52 – ( 2 x 12.2 x 17.5 x cos 137o )
a2 = 148.84 + 306.25 – ( 427 x – 0.731 )Notice the two negative
signs.a2 = 455.09 + 312.137
a2 = 767.227
a = M = 27.7m
Using The Cosine Using The Cosine RuleRule
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Demo
What Goes In The What Goes In The Box ?Box ?
Find the length of the unknown side in the triangles:
(1)78o
43cm
31cmL
(2)
8m
5.2m
38o
M
L = 47.5cm
M =5.05m
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2
o o
1. I f lines have the same gradient
What is special about them.
2. Factorise x +4x -12
3. Explain why the missing angles
are 90 and 36
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54o
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1.1. Know when to use the Know when to use the cosine rule to solve cosine rule to solve REAL LIFE problems.problems.
1. We are learning when to use the cosine rule to solve REAL LIFE problems involving finding an angle of a triangle .
Cosine RuleCosine RuleNat 5
2. 2. Solve Solve REAL LIFE problems problems that involve finding an that involve finding an angle of a triangle.angle of a triangle.
C
B
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a
b
c
The Cosine Rule can be used with ANY triangle as long as we have been given enough information.
Works for any Triangle
cos2 2 2a =b +c - 2bc A
Finding Angles Finding Angles Using The Cosine RuleUsing The Cosine Rule
Consider the Cosine Rule again:We are going to change the subject of the formula to cos Ao
Turn the formula around:b2 + c2 – 2bc cos Ao = a2
Take b2 and c2 across.-2bc cos Ao = a2 – b2 – c2
Divide by – 2 bc.2 2 2
cos2
o a b cA
bc
Divide top and bottom by -12 2 2
cos2
o b c aA
bc
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Write down the formula for cos Ao
2 2 2
cos2
o b c aA
bc
Label and identify Ao and a , b and c.
Ao = ? a = 11b = 9 c = 16
Substitute values into the formula.
2 2 29 16 11cos
2 9 16oA
Calculate cos Ao .Cos Ao =0.75
Use cos-1 0.75 to find Ao
Ao = 41.4o
Example 1 : Calculate the
unknown angle Ao .
Finding Angles Finding Angles Using The Cosine RuleUsing The Cosine Rule
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Demo
Example 2: Find the unknown
Angle yo in the triangle:
Write down the formula.
2 2 2
cos2
o b c aA
bc
Identify the sides and angle.
Ao = yo a = 26 b = 15 c = 13
2 2 215 13 26cos
2 15 13oA
Find the value of cosAo
cosAo = - 0.723The negative tells you the angle is obtuse.
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Finding Angles Finding Angles Using The Cosine RuleUsing The Cosine Rule
Works for any Triangle
Demo
What Goes In The Box ?What Goes In The Box ?
Calculate the unknown angles in the triangles below:
(1)
10m
7m5m Ao
Bo
(2) 12.7c
m
7.9cm
8.3cm
Ao =111.8o
Bo = 37.3o
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2
1. A washing machine is reduced by 10%
in a sale. I t's sale price is £ 360.
What was the original price.
2. Factorise x - 7x +12
3. Find the missing angles.
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61o
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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.
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