Super Trig PowerPoint
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Transcript of Super Trig PowerPoint
Super Trig PowerPoint
Introduction- Finding lengths
Sin, cos or tan to find lengths?
Finding missing lengths worksheet
Finding the Area of Triangles and
Segments
The Sine rule The Cosine RuleThe Sine and
Cosine Rule (quiz and worksheet)
Finding lengths and angles (more
practise)
Finding Missing Angles
The Sine rule proof
The Cosine Rule proof
Trig graphs Combining the Rules
Trig graphs- matching cards
a
b
c
d
e
f
g h
i10cm30˚
40˚50˚
35˚
45˚ 42˚
27˚
38˚
51˚
Use sin, cos and tan to find the missing lengths, round them to 1 d.p, and use that
answer to work out the next length.Side Length (rounded to 1 dp)
a
b
c
d
e
f
g
h
i
a
b
c
d
e
f
g h
i10cm30˚
40˚50˚
35˚
45˚ 42˚
27˚
38˚
51˚
Use sin, cos and tan to find the missing lengths, round them to 1 d.p, and use that
answer to work out the next length.Side Length (rounded to 1 dp)
a
b
c
d
e
f
g
h
i
Side Length (rounded to 1 dp)
a 5
b 4.2
c 3.2
d 2.2
e 3.1
f 3.4
g 1.7
h 2.8
i 9.5
Home
Trigonometry 1
Warm upSolve the following
equations:
1) 20=
2) 15=
3) 8=
4) 7=
5) 16=
X2
X3
32 X
21 X
64 X
Trigonometry• We can use trigonometry to find missing
angles and lengths of triangles.• Trigonometry uses three functions, these are
called:– Sine (shortened to Sin and pronounced “sign”)– Cosine (shortened to Cos)– Tangent (shortened to Tan)
• We will start working with right angled triangles
Labelling the sides
Hypotenuse
The longest side, the one opposite the right angle is called the hypotenuse
Before we can use Sin, Cos and Tan we need to be able to label the sides of a right angled triangle
Labelling the sidesO
ppos
iteWhat we call the other two sides will
change depending on which angle we are working with, for example..
ϴAdjacent
If we are given (or need to work out) this angle, we label the other
sides like this..But if we are working
with this angle, we label the sides like
this...
Opposite
Adja
cent
Labelling Right Angle Triangle
10 multiple choice questions
OppositeAdjacent
Hypotenuse
A) B)
C)
ϴX
What is the side marked with an X?
Adjacent
Hypotenuse OppositeA) B)
C)
ϴ
X
What is the side marked with an X?
Hypotenuse Opposite
Adjacent
A) B)
C)
ϴ
X
What is the side marked with an X?
Opposite Adjacent
Hypotenuse
A) B)
C)
ϴ
X
What is the side marked with an X?
Hypotenuse
Adjacent OppositeA) B)
C)
ϴX
What is the side marked with an X?
AdjacentOpposite
Hypotenuse
A) B)
C)
ϴX
What is the side marked with an X?
AdjacentOpposite
Hypotenuse
A) B)
C)
ϴ
X
What is the side marked with an X?
Opposite Hypotenuse
Adjacent
A) B)
C)
ϴX
What is the side marked with an X?
Opposite
Hypotenuse AdjacentA) B)
C)
ϴ
X
What is the side marked with an X?
Hypotenuse Opposite
Adjacent
A) B)
C)
ϴ
X
What is the side marked with an X?
Sine (sin)
30˚
5cm10cm We use Sine when we have the
Opposite length and the Hypotenuse
Try entering sin30 in your calculator, it should give the same answer as 5 ÷ 10
Sin30= 510
The rule we use is:
Sinϴ= OppositeHypotenuse
Sin Example 1
42˚
O 7cm We can use Sin as the question involves the Opposite length and the Hypotenuse
Sin42= O7
The rule we use is:
Sinϴ= OppositeHypotenuse
7 x Sin42= O
4.68 cm (2dp)= O
Sin Example 2
17˚
10cm H We can use Sin as the question involves the
Opposite length and the Hypotenuse
Sin17= 10H
The rule we use is:
Sinϴ= OppositeHypotenuse
H x Sin17= 10
H= 10 Sin17 H= 34.2 cm (1dp)
Cosine (cos)
50˚
Adjacent
Hypotenuse
We use cosine when we have the Adjacent length and the Hypotenuse
The rule we use is:
Cosϴ= AdjacentHypotenuse
Cos Example 1
53˚
A
9cm We can use Cos as the question involves the Adjacent length and the Hypotenuse
Cos53= A9
The rule we use is:
Cosϴ= AdjacentHypotenuse
9 x Cos53= A
5.42 cm (2dp)= A
Cos Example 217˚
9cm H We can use Cos as the question involves
the Adjacent length and the Hypotenuse
Cos17= 9H
The rule we use is:
Cosϴ= AdjacentHypotenuse
H x Cos17= 10
H= 9 Cos17 H= 9.41 cm (2dp)
Tangent (tan)
50˚
6.4cm (1dp)
10cm
We use tangent when we have the Opposite and Adjacent lengths.
The rule we use is:
Tanϴ= OppositeAdjacent
Tan Example 1
53˚
O
11cm
We can use Tan as the question involves the Adjacent and Opposite lengths
Tan53= O11
The rule we use is:
Tanϴ= OppositeAdjacent
11 x Tan53= O
14.6 cm (1dp)= O
Tan Example 2
35˚
A
21cm
We can use Tan as the question involves the Adjacent and Opposite lengths
Tan35= 21A
The rule we use is:
Tanϴ= OppositeAdjacent
A x Tan35= 21
A= 21 Tan35
A= 29.99 cm (2dp)
The three rulesSo we have:
Tanϴ= OppositeAdjacent
Cosϴ= AdjacentHypotenuse
Sinϴ= OppositeHypotenuse
Sinϴ= Tanϴ= OA
Cosϴ= AH
OH
SOHCAHTOAThere are a few ways to remember this
Silly Old Horses Can’t Always Hear The Other Animals
SOHCAHTOA
WHAT?
Practise1. Use Sine to find the missing lengths on these triangles:
2. Use Cosine to find the missing lengths on these triangles:
3. Use Tangent to find the missing lengths on these triangles:
O15cm
50˚17cm
H60˚
A
22cm
38˚
25cmH60˚
O
15cm42˚
11cm
A60˚ Tanϴ=Opposite
Adjacent
Cosϴ= AdjacentHypotenuse
Sinϴ=Opposite
Hypotenuse
Home
Trigonometry 2
Skiers On Holiday Can Always Have The Occasional Accident
SOHCAHTOA
Tanϴ= OppositeAdjacent
Cosϴ= AdjacentHypotenuse
Sinϴ= OppositeHypotenuse
Our aim today• We have looked at the three rules and have
practised labelling triangles.• Today we will have to decide whether we are
using Sin, Cos or Tan when answering questions.
SOH CAH TOA
7cmX
35˚oppo
site
Hypotenuse
This question will use Sine
Sin35= X7
Sinϴ= OH
SOH CAH TOA
X
8cm
17˚
opposite
Adja
cent
This question will use Tan
Tan17= 8X
Tanϴ= OA
SOH CAH TOA
X
8cm
43˚
opposite
Hypotenuse
This question will use Sin
Sin43= 8X
Sinϴ= OH
SOH CAH TOA
X
8cm26˚
Hypotenuse
Adja
cent
This question will use Cosine
cos26= X8
cosϴ= OA
Sin, Cos or Tan?
10 multiple choice questions
SinCos
Tan
A) B)
C)
35˚X
Will you use Sin, Cos or Tan with this question?
11cm
Cos
Sin TanA) B)
C)
14˚X
Will you use Sin, Cos or Tan with this question?
15cm
Sin Cos
Tan
A) B)
C)
40˚ XWill you use Sin, Cos or Tan with this question?
17cm
Tan Sin
Cos
A) B)
C)
50˚
X
Will you use Sin, Cos or Tan with this question?
5cm
Sin
Cos TanA) B)
C)
51˚X
Will you use Sin, Cos or Tan with this question?
6cm
TanSin
Cos
A) B)
C)
16˚X
Will you use Sin, Cos or Tan with this question?
8cm
CosSin
Tan
A) B)
C)
42˚14cm
Will you use Sin, Cos or Tan with this question?
X
Tan Cos
Sin
A) B)
C)
35˚X
Will you use Sin, Cos or Tan with this question?
4cm
Sin
Cos TanA) B)
C)
63˚
X
Will you use Sin, Cos or Tan with this question?
3.4cm
Sin Tan
Cos
A) B)
C)
71˚
X
Will you use Sin, Cos or Tan with this question?
5mm
Answers:1) 3.1cm2) 6.1cm3) 5.1cm4) 17.1cm5) 4.5cm6) 8.6cm7) 20.5cm8) 31.1cm9) 117.6cm10)1.5cm11)4.1cm12)108.9cm
Practise
Home
The Sine Rule
a
b c
C
A
B
SinA SinB SinC==a b c
This is the Sine rule
We can also write it like this
We can use it to find out missing sides and angles of non right angle triangles
7cm
5cm 50˚
B
Because we’re looking for an angle, I'm going to use the version of the rule which has Sin on top
SinA SinB=a b
Sin50 SinB=7 5
(multiply both sides by 5)
5xSin50 SinB=7
SinB=0.54717...
Sin-10.54717... = 33.2˚ (1dp)
11cm
8cm38˚
C
Because we’re looking for an angle, I'm going to use the version of the rule which has Sin on top
SinA SinB=a b
Sin38 SinC=11 8
(multiply both sides by 58)
8xSin38 SinC=11
SinC=0.447753..
Sin-10.447753... =26.6˚ (1dp)
a
5cm 50˚
35˚
Because we’re looking for a length, I'm going to use the version of the rule which has the length on top
a b=sinA sinB
Sin35 Sin50=5 a (multiply both sides by
Sin50)
5xSin50 a=sin35
a=6.7cm(1dp)
c
9cm
64˚
21˚
Because we’re looking for a length, I'm going to use the version of the rule which has the length on top
a c=SinA SinC
Sin64 Sin21=9 c (multiply both sides by
Sin21)
9xSin21 c=sin64
c=3.6cm(1dp)
The Sine Rule Quiz
10 multiple choice questions
Yes NoA) B)
Can you use the sine rule to find the missing value?
10cm
12cm
a
20˚
NoYesA) B)
Can you use the sine rule to find the missing value?
9cm
15cm
a
30˚
Yes NoA) B)
Can you use the sine rule to find the missing value?
10cm
X20˚
85˚
NoYesA) B)
Can you use the sine rule to find the missing value?
10cm 12cm
a
15cm
Yes NoA) B)
Can you use the sine rule to find the missing value?
10cm
12cm
38 ˚
20˚
X
NoYesA) B)
Can you use the sine rule to find the missing value?
a
20˚
72˚ 88˚
NoYesA) B)
Can you use the sine rule to find the missing value?
10cm
12cm
Yes NoA) B)
Can you use the sine rule to find the missing value?
10cm
x35˚
But you could also useSin35= 10 ÷ x
Yes NoA) B)
Can you use the sine rule to find the missing value?
9.8cm
21cm
a
34˚
Yes NoA) B)
Can you use the sine rule to find the missing value?
5cm
7cm
a
6cm
Practise Questions
37˚
45˚
61˚45˚
23˚
37˚
80˚
a
b
19˚
110˚c
11cm
BC 8cm
25cm
C
5.6cm
7.2cm
59cm
47cm57cm
34cm
Answers:1) 9.4cm2) 9.9cm3) 38.5cm4) 50˚5) 15˚6) 34.1˚
Home
The Cosine Rule
Warm up
7cm
x
x23cm
19cm29cm 7.1cm
13.4cm
35˚ 41˚
ϴ
ϴ
x= 8.5cmx= 15.1cm
ϴ=49.1˚
ϴ=27.9˚
Find the missing sides and angles
a
Abc
a2=b2 + c2 – 2bccosA
The length here, has to be the length opposite this angle
Which side is b and which is c doesn’t matter
a
95˚5cm8cm
a2=52 + 82 – 2x5x11xcos95
The length here, has to be the length opposite this anglea2=25 + 64 – 110xcos95a2=89– 110xcos95a2=89– -9.58.....a2=98.58...
a=9.9291..a=9.9cm (1dp)
a
81˚7cm12cm
a2=72 + 122 – 2x7x12xcos81
The length here, has to be the length opposite this anglea2=49 + 144 – 168xcos81a2=193– 168xcos81a2=193– 26.280...a2=166.7190...
a=12.911..a=12.9cm (1dp)
11cm
A5cm8cm
112=52 + 82 – 2x5x11xcosA
The length here, has to be the length opposite this angle
112=52 + 82 – 2x5x11xcosA121=89– 2x5x11xcosA32=-110xcosA32 ÷ -110=cosA
-0.2909090..=cosACos-1-0.209090..=A106.9˚ (1dp)=A
20cm
A13cm8cm
202=132 + 82 – 2x13x11xcosA
The length here, has to be the length opposite this angle
202=132 + 82 – 2x13x8xcosA400=233– 208cosA167=-208xcosA167 ÷ -208=cosA
-0.80288.....cosACos-1-0.80288...=A143.4˚ (1dp)=A
Practise Questions
12cm
45˚ 61˚
15cm
28cm
37˚
3.4cm
a
b
51cm
23cm
c
11cm
xx 8cm
25cm
C
5.6cm
7.2cm
59cm
47cm57cm
48cm
Answers:1) 1.8.9cm2) 13.1cm3) 17.1cm4) 49.1˚5) 49.9˚6) 55.8 ˚
Home
The Sine Rule proof
What would you like to do?
Be shown the proof Try to prove it yourself
Why does the Sine Rule Work?A
B
bc
aB
Using the rule that:Sinϴ=O/AWe can show that the red line is: Red line=c x sinB
b
C
Using the rule that:Sinϴ=O/AWe can show that the red line is: Red line=b x sinC
The red line is b x sinC and c x sinBSo b x sinC= c x sinB
We can rearrange to make:
bsinB =
csinC
SinBb =
SinCc
or
Why does the Sine Rule Work?A
B
bc
aB
Using the rule that:Sinϴ=O/AWe can show that the red line is: Red line=a x sinB
b
C
Using the rule that:Sinϴ=O/AWe can show that the red line is: Red line=b x sinA
The red line is b x sinA and a x sinBSo b x sinA= a x sinB
We can rearrange to make:
bsinB =
asinA
A
a
SinBb =
SinAa
or
Why does the Sine Rule Work?
So far we have shown that:
bsinB =
asinA
bsinB =
csinCand
Therefore:a
sinA =b
sinB =c
sinC
(We could have also used the version with the angles on top)
Home
The Sine Rule
What would you like to do?
Be shown the proof Try to prove it yourself
Prove the Sine Rule!
Hint 1Start with a diagram like this:
CA
B
b
ac
Hint 2
Hint 3
Split it into 2 right angled triangles
Can you find 2 different ways to find the length of the line you drew?
Hint 4Can you split the triangle any other way?
Show me the proof Home
Finding the Areas of Triangles
40˚
10cm
8cm h8cm h
40˚Can we find the height of this triangle using trigonometry?
Sin40=h÷8
8 x Sin40=hSo the height of the triangle is 8 x sin40, we know that the area of a triangle is half base times height so..
Area= ½ base x height
Area= ½ x 10 x 8 x sin40
Can we find the area of this triangle?We need the base and the height (area= half base times height)
a
hb h
CCan we find the height of this triangle using trigonometry?
SinC=h÷b
b x SinC=hSo the height of the triangle is b x sinC, we know that the area of a triangle is half base times height so..
Area= ½ base x height
Area= ½ x a x b x sinC
Can we find the area of this triangle?This is what we’d call
The formula for the area of a triangle is:
Area= ½ absinC
The formula for the area of a triangle is:
Area= ½ absinCWe can use this formula to find the area of non right angled triangles when we haven’t been given the perpendicular height
All we need to know is:The length of two sides and the size of the angle between them
Practise Questions Challenge Questions
Area=30cm2
Area=30cm2
Area=25m2
Area=75cm2
Area=52cm2
Find the areas of the following triangles Find the missing lengths/angles of the following triangles
10cm
40˚
8cm 12cm
9cm
12˚ 72˚
7cm
6.5cm
19cm 9.1cm
ϴ
20cm
13cm
35˚
ϴ15cm
12cm
14cm11cm
38˚7cm a
a
10cm28˚
28˚
13cm
bAnswers:1) 28.9cm2
2) 21.6cm2
3) 10cm2
4) 49.6cm2
5) 70.8cm2
Answers:1) 13.9cm2) 21.3cm3) 9.8cm4) 56.4˚5) 42.5 ˚
Area of Segments
Here we will look at finding the area of sectors
You will need to be able to do two things:
Area of Segments
1) Find the area of a sector using the formula-
2) Find the area of a triangle using the formula-
Area= ½ absinC
Area of sector= Angle of Sector x πr2
360
C
b
a
Example-find the area of the blue segment
10cm10cm100°
Step 1- find the area of the whole sectorArea= 100/360 x π x r2
= 100/360 x π x 102
=100/360 x π x 100 =87.3cm2
Step 2- find the area of the triangleArea= ½ absinC =1/2 x 10 x 10 x sin100 = 49.2cm2
Step 3- take the area of the triangle from the area of the segment
87.3 – 49.3 = 38 cm2
Example-find the area of the blue segment
12cm12cm120°
Step 1- find the area of the whole sectorArea= 120/360 x π x r2
= 120/360 x π x 122
=120/360 x π x 144 =150.8cm2
Step 2- find the area of the triangleArea= ½ absinC =1/2 x 12 x 12 x sin120 = 62.4cm2
Step 3- take the area of the triangle from the area of the segment
150.8 – 62.4 = 88.4 cm2
Questions
10cm
130°11cm85°
12cm170°
5cm95°6.5cm
Find the area of the blue segments, to 1 decimal place
1 2 3
654
17cm65°160°
Answers:1) 75.1cm2
2) 29.5cm2
3) 201.1cm2
4) 8.3cm2
5) 51.8cm2
6) 33cm2
Home
Trigonometry 3Finding missing angles
Some Old Hairy Camels Are Hairier Than Other Animals
SOHCAHTOA
Tanϴ= OppositeAdjacent
Cosϴ= AdjacentHypotenuse
Sinϴ= OppositeHypotenuse
• Alan presses SIN then an angle, he gets the answer 0.5, what angle did he enter?
Sinϴ=0.5We can use the inverse of sin to find out
Sin-10.5= ϴ30˚= ϴ
Warm up
Warm upUse the inverse of Sin, Cos and Tan to find the missing
angles, rounded to 1dp:1. sinϴ=0.7 (type sin-10.7)
2. sinϴ=0.33. cosϴ=0.54. cosϴ=0.95. tanϴ=0.36. tanϴ=0.25
Answers:1)44.4˚2)17.5˚3)60˚4)25.8˚5)16.7˚6)14˚
SOH CAH TOA
7cm3cm
ϴoppo
site
Hypotenuse
This question will use Sin
Sinϴ= 37
Sinϴ= OH
Find the missing angle
Sinϴ=0.42857...What angle would give us this
answer?
Sin-10.42857...= ϴ25.4˚ (1dp)= ϴ
You could use the ANS button on your calculator
Sin-1ANS= ϴ
SOH CAH TOA
8cm
6cm ϴ
Adja
cent
Opposite
This question will use Tan
Tanϴ= 86
Tanϴ= OA
Find the missing angle
Tanϴ=1.25What angle would give us this
answer?
Tan-11.25= ϴ51.3˚ (1dp)= ϴ
SOH CAH TOA
12cm9cm ϴ
Adja
cent
Hypotenuse
This question will use Cos
Cosϴ= 912
Cosϴ= AH
Find the missing angle
Cosϴ=0.75What angle would give us this
answer?
Cos-10.75= ϴ41.4˚ (1dp)= ϴ
You could use the ANS button on your calculator
Cos-1ANS= ϴ
Practise Questions
ϴ
ϴ
ϴ
ϴ
ϴ
ϴ
ϴϴ
10cm 11
cm
17cm
11cm
20cm
12cm 18cm
5cm 15cm
23cm 35cm
22cm
10cm
12cm
6cm
13cmAnswers:1) 50.2˚2) 28.6˚3) 59.1˚4) 52.3˚5) 63.4˚6) 65.4˚7) 28.6˚8) 40.9˚
Home
The Cosine Rule
What would you like to do?
Be shown the proof Try to prove it yourself
Prove the Cosine Rule!
Hint 1Start with a diagram like this:
A
b
ac
Hint 2
Hint 3
Split it into 2 right angled triangles
You could use pythagoras to find the length of a in the right angled triangle on the right if you had the other two lengths
Hint 4Expand and tidy up, using factorising (don’t forget the identity at the top!)
Show me the proof Home
For this proof you need to know that- (SinA)2 + (CosA)2 = 1
Home
Find the missing lengths and angles
X16cm
30˚
1
14cm
X
51˚
2
15cm
17cm
ϴ3
8cm12cm
ϴ
4
19cm36cm
ϴ5
X18cm
63˚
6
9cm
8.3cm
ϴ7
11.2cm X
35˚
8
X
15cm
43˚
9
X
23cm
50˚10
40cm53cmϴ
X
46cm
28˚
16cm32cmϴ
X
36cm
18˚
61cm 74cm
ϴ 81cm
106cmϴ
11 12
16151413
Answers:1) 8cm2) 22.2cm3) 48.6 ˚4) 41.8 ˚5) 58.1 ˚6) 16cm7) 42.7 ˚8) 19.5cm9) 11cm10)19.3˚11)41 ˚12)21.6cm13)60 ˚14)11.7cm15)55.5 ˚16)49.8 ˚
Home
Why does the Cosine rule work?
a
a
c
A
The cosine rule has “a” as it’s subject, so we need to think of how we could find a in this triangleIf we knew the lengths of the red and purple lines, we could use Pythagoras’ theorem to find aThe red is cSinA
The purple will be the length of b takeaway the green length
The green will be cCosA so the purple is b-cCosA
b-cCosA
cSinAa
From Pythagoras we know that:
a2= (cSinA)2 + (b-cCosA)2
a2= c2(SinA)2 + b2+c2(CosA)2 –bcCosA-bcCosA
a2= b2+c2(SinA)2 +c2(CosA)2 –2bcCosA
a2= b2+c2((SinA)2 + (CosA)2) –2bcCosA
a2= b2+c2(1)–2bcCosA
a2= b2+c2–2bcCosA
So.....
(SinA)2 + (CosA)2 = 1This is something that will come up whilst you are doing your A levels
QED
QED stands for the latin phrase “quod erat demonstrandum” which mean, “which had to be demonstrated” it’s a way for us to say we have finished our proof
Home
Sine or Cosine Rule?
10 multiple choice questions
Sine Rule Cosine RuleA) B)
How would you find the missing value?
X
11cm
120˚
11˚
Cosine RuleSine RuleA) B)
How would you find the missing value?
X
62cm40cm
110cm
Sine Rule Cosine RuleA) B)
How would you find the missing value?
X
11cm
86˚25˚
Cosine RuleSine RuleA) B)
How would you find the missing value?
X
16cm40cm
32cm
Sine Rule Cosine RuleA) B)
How would you find the missing value?
X
6cm22˚ 110˚
Cosine RuleSine RuleA) B)
How would you find the missing value?
X
6cm
4cm110˚
Cosine RuleSine RuleA) B)
How would you find the missing value?
X
3.7cm
4.2cm 81˚
Sine Rule Cosine RuleA) B)
How would you find the missing value?
X
6cm32˚
115˚
Sine Rule Cosine RuleA) B)
How would you find the missing value?
X
13cm
27cm
114˚
Cosine RuleSine RuleA) B)
How would you find the missing value?
X6cm
4cm
7cm
Find the missing lengths and angles(give your answers to 1dp)
13cm
9.2cm9cm
ϴ
ϴ
8cm
7cm
5cm
11cm
12cm
7cm
ϴ
13cm
11cm35˚
ϴ
ϴ
17cm
22cm
41˚
45˚
17˚ 25cm
X
25cm X77˚
˚32˚
X
14cm11cm 43˚
X14cm
23cm31˚
Answers:1) 91.2˚2)29 ˚3)10.3cm4)9.6cm5)60 ˚6)30.5 ˚7)13.2cm8)27.9cm9)35.1 ˚
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For each question say what method you would use to find the missing
value
Warm up
Combining the Rules
Pythagoras’ Theorem
Sine Rule
Cosine Rule
SOHCAHTOAA) B)
C) D)
How would you find the missing length in this triangle?
7cm
6cm
x
SOHCAHTOASine Rule
Cosine RulePythagoras’ Theorem
A) B)
C) D)
How would you find the missing length in this triangle?
12cm x
30˚
Cosine Rule
Sine Rule SOHCAHTOA
Pythagoras’ Theorem
A) B)
C) D)
How would you find the missing length in this triangle?
13cmx
46˚ 19cm
SOHCAHTOASine Rule
Cosine RulePythagoras’ Theorem
A) B)
C) D)
How would you find the missing length in this triangle?
13cm
x40˚
Sine Rule SOHCAHTOA
Cosine RulePythagoras’ Theorem
A) B)
C) D)
How would you find the missing length in this triangle?
X
11cm
120˚
11˚
Cosine Rule
Sine Rule SOHCAHTOA
Pythagoras’ Theorem
A) B)
C) D)
How would you find the missing length in this triangle?
X
6cm
4cm110˚
SOHCAHTOASine Rule
Cosine RulePythagoras’ Theorem
A) B)
C) D)
How would you find the missing length in this triangle?
12cm
x25˚
Cosine Rule
Sine Rule SOHCAHTOA
Pythagoras’ Theorem
A) B)
C) D)
How would you find the missing length in this triangle?X
3.7cm
4.2cm 81˚
Pythagoras’ Theorem
Sine Rule
Cosine Rule
SOHCAHTOAA) B)
C) D)
8cm16cm
x
Sine Rule SOHCAHTOA
Cosine RulePythagoras’ Theorem
A) B)
C) D)
How would you find the missing length in this triangle?
X
6cm22˚ 110˚
Work out the missing lengths (give answers to 1dp)
X 14cm
23cm
71˚
9cm
32˚130˚
c
a
b
d
Circumference=100cm
81˚
X
4cm
15cm
15cm
23˚
X
28˚
Area of blue segment- 200cm2
X
120˚
146˚
9cm12cm
X
Answers:1) 23.4cm2)
a) 4.8cmb) 7.6cmc) 14.3cmd) 11.3cm
3) 20.7cm4) 24.7cm5) 18.4cm6) 9.6cm
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Graphs of Trig Functions
The Sine Curve
The Cosine Curve
The Tan Curve
Key ValuesAngle SinX CosX
0 0 190 1 0
180 -0 -1270 -1 0360 0 1
As you can see both sine and cosine follow the same pattern.
The difference is that sine starts at 0, whereas cosine starts at 1
Key ValuesAngle TanX
0 090 ∞
180 0270 ∞360 0
Reading off values
If sinx=0.5 what is x?A calculator would tell us that it is 30˚, but there are other values that would give 0.5
30 150
The solutions for sinx=0.5 between 0 and 360 are x=30 and x=150
-0.5
Reading off values
If sinx=-0.3 what is x?
197.5 342.5
The solutions for sinx=-0.3 between 0 and 360 are x=197.5 and x=342.5
-0.3
Transformation of trig graphs• Here instead we will use f(x) instead of sin, cos or tan, as what we will
describe works not only with these three functions but all other functions.• y=f(x) + k Translation through
• y=f(x+k) Translation through
• y=kf(x) Stretch by factor k parallel to the y axis
• y=f(kx) Stretch by factor 1/k parallel to the x axis
• y=-f(x) Reflection in the x axis
• y=f(-x) Reflection in the y axis
-k0 )(
0k )(
y=sinxy=2sinx (stretches the curve vertically)y=0.5sinx (squashes vertically)
y=sinxy=sin0.5x (stretches horizontally)
y=sin2x (squashes horizontally)
y=sinxy=sinx + 1 (moves curve upwards)y=sin(x+90) (moves curve to the right)
Transformations of Trig Graphs
10 multiple choice questions
Y=sinx
Y=cosx
Y=cosx-1
Y=tanxA) B)
C) D)
Y=cosxY=tanx
Y=cos2xY=2cosx
A) B)
C) D)
Y=tanx
Y=tan2x Y=0.5tanx
Y=-tanx
A) B)
C) D)
Y=2cosx Y=cos2x
Y=2sinxY=sin2x
A) B)
C) D)
Y=0.5sinx
Y=sin2x y=sin0.5x
Y=0.5cosx
A) B)
C) D)
Y=-cosxY=-sinx
Y=sinx-1Y=cos(-x)
A) B)
C) D)
Y=sin2xY=0.5sinx
Y=2sinxY=sin0.5x
A) B)
C) D)
Y=tanx-1
Y=tanx+1 Y=tan(x-1)
Y=tanx
A) B)
C) D)
Y=-tanx
Y=tan(x+90)
Y=tan2x
Y=2tanxA) B)
C) D)
Y=cosx-1 Y=-cosx
Y=cos(-x)Y=sinx-1
A) B)
C) D)
Practise Questions
1. Sketch the graph y=cosxa) Use your graph to find all of the solutions for cosx=0.7 between 0 and
360b) Use your graph to find all of the solutions for cosx=-0.5 between 0 and
360
2. Sketch the graph y=tanxa) Use your graph to find all the solutions for tanx=1.5 between 0 and 360
3. Sketch the following graphs:b) y=3sinxc) y=sin3xd) y=cosx + 5e) y=cos(x-45)f) Y=tan(x+10)
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y=sinxy=cosxy=tanx
y=2cosx
y=sin2xy=-cosx
y=sin(x-90)y=sin(x+90)
y=0.5sinx y=tanx-1y=tan(-x) y=cosx-1y=-tanx y=cos0.5x
Cut out the cards and match them to the graphs- (some graphs may have more than one label!)
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