Trigonometry

1
Trigonometry
S5 Int2
Exact Values
Angles greater than 90o
Useful Notation Area of a triangle
Using Area of Triangle Formula
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Sine Rule Problems
Cosine Rule Problems
Mixed Problems
2
Starter Questions
S5 Int2
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3
Exact Values
S5 Int2
Learning Intention
Success Criteria

1. Recognise basic triangles and exact values for
   sin, cos and tan 30o, 45o, 60o .

1. To build on basic trigonometry values.

1. Calculate exact values for problems.

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4
Exact Values
S5 Int2
Some special values of Sin, Cos and Tan are
useful left as fractions, We call these exact
values
30º
?3
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1
This triangle will provide exact values for sin,
cos and tan 30º and 60º
5
Exact Values
S5 Int2
x 0º 30º 45º 60º 90º
Sin xº
Cos xº
Tan xº ?
?3 2
½
1
0
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?3 2
1
½
0
0
?3
6
Exact Values
S5 Int2
Consider the square with sides 1 unit
45º
?2
1
1
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45º
1
1
We are now in a position to calculate exact
values for sin, cos and tan of 45o
7
Exact Values
S5 Int2
x 0º 30º 45º 60º 90º
Sin xº
Cos xº
Tan xº ?
?3 2
1 ?2
½
1
0
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?3 2
1 ?2
1
½
0
0
1
?3
8
Exact Values
S5 Int2
Now try Exercise 1 Ch8 (page 94)
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9
Starter Questions
S5 Int2
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10
Angles Greater than 90o
S5 Int2
Learning Intention
Success Criteria

1. Find values of sine, cosine and tangent over the
   range 0o to 360o.

1. Introduce definition of sine, cosine and tangent
   over 360o using triangles with the unity circle.

2. Recognise the symmetry and equal values for
sine, cosine and tangent.
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11
Angles Greater than 90o
S5 Int2
We will now use a new definition to cater for ALL
angles.
New Definitions
y-axis
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r
Ao
x-axis
O
12
Trigonometry
Angles over 900
S5 Int2
Example 1
The radius line is 2cm. The point (1.2,
1.6). Find sin cos and tan for the angle.
(1.2, 1.6)
Check answer with calculator
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53o
13
Trigonometry
Angles over 900
S5 Int2
Example 1
Check answer with calculator
The radius line is 2cm. The point (-1.8,
0.8). Find sin cos and tan for the angle.
(-1.8, 0.8)
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127o
14
What Goes In The Box ?
S5 Int2
Write down the equivalent values of the following
in term of the first quadrant (between 0o and
90o)

1. Sin 300o
2. Cos 360o
3. Tan 330o
4. Sin 380o
5. Cos 460o

1. Sin 135o
2. Cos 150o
3. Tan 135o
4. Sin 225o
5. Cos 270o

- sin 60o
sin 45o
-cos 45o
cos 0o
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- tan 30o
-tan 45o
sin 20o
-sin 45o
- cos 80o
-cos 90o
15
Trigonometry
Angles over 900
S5 Int2
Now try Exercise 2 Ch8 (page 97)
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16
Trigonometry
Angles over 900
S5 Int2
Extension for unit 2 Trigonometry
GSM Software
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17
Angles Greater than 90o
Int 2
Two diagrams display same data in a different
format
Sin ve
(0,1)
All ve
180o - xo
(1,0)
(-1,0)
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360o - xo
180o xo
(0,-1)
Cos ve
Tan ve
18
Starter Questions
S5 Int2
3cm
8cm
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19
Area of a Triangle
S5 Int2
Learning Intention
Success Criteria

1. Be able to label a triangle properly.

• To show the standard way of labelling a triangle.
• 2. Find the area of a triangle using basic
  trigonometry knowledge.

2. Find the area of a triangle using basic
trigonometry knowledge.
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20
Labelling Triangles
S5 Int2
In Mathematics we have a convention for labelling
triangles.
B
B
a
c
C
C
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b
A
A
Small letters a, b, c refer to distances
Capital letters A, B, C refer to angles
21
Labelling Triangles
S5 Int2
Have a go at labelling the following triangle.
E
E
d
f
F
F
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e
D
D
22
Area of a Triangle
S5 Int2
Example 1 Find the area of the triangle ABC.
B
(i) Draw in a line from B to AC
(ii) Calculate height BD
10cm
7.66cm
50o
D
(iii) Area
C
A
12cm
23
Area of a Triangle
S5 Int2
Example 2 Find the area of the triangle PQR.
P
(i) Draw in a line from P to QR
(ii) Calculate height PS
12cm
7.71cm
40o
S
(iii) Area
R
Q
20cm
24
Constructing Pie Charts
S5 Int2
Now try Exercise 3 Ch8 (page 99)
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25
Starter Questions
S5 Int2
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26
Area of ANY Triangle
S5 Int2
Learning Intention
Success Criteria

1. Know the formula for the area of any triangle.

1. To explain how to use the Area formula for
ANY triangle.
2. Use formula to find area of any triangle given
two length and angle in between.
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27
General Formula forArea of ANY Triangle
Consider the triangle below
Area ½ x base x height
What does the sine of Ao equal
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Change the subject to h.
h b sinAo
Substitute into the area formula
28
Area of ANY Triangle
Key feature To find the area you need to
knowing 2 sides and the angle in between (SAS)
S5 Int2
The area of ANY triangle can be found by the
following formula.
B
B
a
Another version
C
c
C
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Another version
b
A
A
29
Area of ANY Triangle
S5 Int2
Example Find the area of the triangle.
The version we use is
B
B
20cm
C
c
C
30o
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25cm
A
A
30
Area of ANY Triangle
S5 Int2
Example Find the area of the triangle.
The version we use is
E
10cm
60o
8cm
F
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D
31
What Goes In The Box ?
Key feature Remember (SAS)
S5 Int2
Calculate the areas of the triangles below
A 36.9cm2
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A 16.7m2
32
Area of ANY Triangle
S5 Int2
Now try Exercise 4 Ch8 (page 100)
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33
Starter Questions
S5 Int2
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34
Sine Rule
S5 Int2
Learning Intention
Success Criteria

1. Know how to use the sine rule to solve REAL LIFE
   problems involving 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 .
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35
Sine Rule
Works for any Triangle
S5 Int2
The Sine Rule can be used with ANY triangle as
long as we have been given enough information.
B
a
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c
C
b
A
36
The Sine Rule
Deriving the rule
Draw CP perpendicular to BA
This can be extended to
or equivalently
37
Calculating Sides Using The Sine Rule
S4 Credit
Example 1 Find the length of a in this triangle.
B
C
A
Match up corresponding sides and angles
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Rearrange and solve for a.
38
Calculating Sides Using The Sine Rule
S4 Credit
Example 2 Find the length of d in this triangle.
D
E
C
Match up corresponding sides and angles
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Rearrange and solve for d.
12.14m
39
What goes in the Box ?
S5 Int2
Find the unknown side in each of the triangles
below
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a 6.7cm
b 21.8mm
40
Sine Rule
S5 Int2
Now try Ex 67 Ch8 (page 103)
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41
Starter Questions
S5 Int2
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42
Sine Rule
S5 Int2
Learning Intention
Success Criteria

1. Know how to use the sine rule to solve problems
   involving angles.

1. To show how to use the sine rule to solve
problems involving finding an angle of a
triangle .
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43
Calculating Angles Using The Sine Rule
S4 Credit
B
Example 1 Find the angle Ao
C
Match up corresponding sides and angles
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Rearrange and solve for sin Ao
Use sin-1 0.463 to find Ao
0.463
44
Calculating Angles Using The Sine Rule
S4 Credit
Example 2 Find the angle Xo
Z
Y
Match up corresponding sides and angles
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Rearrange and solve for sin Xo
Use sin-1 0.305 to find Xo
0.305
45
What Goes In The Box ?
S5 Int2
Calculate the unknown angle in the following
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Ao 37.2o
Bo 16o
46
Sine Rule
S5 Int2
Now try Ex 8 9 Ch8 (page 106)
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47
Starter Questions
S5 Int2
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48
Cosine Rule
S5 Int2
Learning Intention
Success Criteria

1. Know when to use the cosine rule to solve
   problems.

1. To show when to use the cosine rule to solve
problems involving finding the length of a side
of a triangle .
2. Solve problems that involve finding the
length of a side.
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49
Cosine Rule
Works for any Triangle
S5 Int2
The Cosine Rule can be used with ANY triangle as
long as we have been given enough information.
B
a
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c
C
b
A
50
Deriving the rule

• 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

Draw BP perpendicular to AC
Since Cos A x/c ? x cCosA
Pythagoras
Pythagoras a bit
Pythagoras - a bit
51
Finding an unknown side.
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
52
Cosine Rule
Works for any Triangle
S5 Int2
How to determine when to use the Cosine Rule.
Two questions
1. Do you know ALL the lengths.
SAS
OR
2. Do you know 2 sides and the angle in between.
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If YES to any of the questions then Cosine Rule
Otherwise use the Sine Rule
53
Using The Cosine Rule
Works for any Triangle
S5 Int2
Example 1 Find the unknown side in the triangle
below
Identify sides a,b,c and angle Ao
a
L
b
5
c
12
Ao
43o
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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.
a L 9.02m
54
Using The Cosine Rule
Works for any Triangle
S5 Int2
Example 2 Find the length of side M.
Identify the sides and angle.
a M
b 12.2
C 17.5
Ao 137o
Write down Cosine Rule
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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
55
What Goes In The Box ?
S5 Int2
Find the length of the unknown side in the
triangles
L 47.5cm
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M 5.05m
56
Cosine Rule
S5 Int2
Now try Ex 12 Ch8 (page 112)
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57
Starter Questions
S5 Int2
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54o
58
Cosine Rule
S5 Int2
Learning Intention
Success Criteria

1. Know when to use the cosine rule to solve REAL
   LIFE problems.

1. To show when to use the cosine rule to solve
REAL LIFE problems involving finding an angle of
a triangle .
2. Solve REAL LIFE problems that involve finding
an angle of a triangle.
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59
Cosine Rule
Works for any Triangle
S5 Int2
The Cosine Rule can be used with ANY triangle as
long as we have been given enough information.
B
a
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c
C
b
A
60
Finding Angles Using The Cosine Rule
Works for any Triangle
S5 Int2
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
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Divide by 2 bc.
Divide top and bottom by -1
You now have a formula for finding an angle if
you know all three sides of the triangle.
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Finding Angles Using The Cosine Rule
Works for any Triangle
S5 Int2
Example 1 Calculate the unknown angle xo .
Write down the formula for cos Ao
a 11
b 9
c 16
Label and identify Ao and a , b and c.
Ao ?
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Substitute values into the formula.
Cos Ao
0.75
Calculate cos Ao .
Use cos-1 0.75 to find Ao
Ao 41.4o
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Finding Angles Using The Cosine Rule
Works for any Triangle
S5 Int2
Example 2 Find the unknown Angle in the
triangle
Write down the formula.
Ao yo
a 26
b 15
c 13
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Identify the sides and angle.
Find the value of cosAo
The negative tells you the angle is obtuse.
cosAo
- 0.723
Ao yo
136.3o
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What Goes In The Box ?
S5 Int2
Calculate the unknown angles in the triangles
below
Bo
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Bo 37.3o
Ao 111.8o
64
Cosine Rule
S5 Int2
Now try Ex 13 Ch8 (page 114)
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65
Starter Questions
S5 Int2
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61o
66
Mixed problems
S5 Int2
Learning Intention
Success Criteria

1. Be able to recognise the correct trigonometric
   formula to use to solve a problem involving
   triangles.

1. To use our knowledge gained so far to solve
various trigonometry problems.
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67
Exam Type Questions
Angle TDA
180 35 145o
Angle DTA
180 170 10o
10o
36.5
SOH CAH TOA
68
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.
• Make a sketch of the journey.
• Find the bearing of the lighthouse from the
  harbour. (nearest degree)

69
Exam Type Questions
Angle ATC
Angle ACT
180 115 65o
180 70 110o
180 110 70o
Angle BCA
65o
110o
70o
53.21 m
SOH CAH TOA
70
Exam Type Questions
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.
71
Mixed Problems
S5 Int2
Now try Ex 14 Ch8 (page 117)
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