Transcript Volume of Prisms

Trigonometry Equations
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National 5
Trig Function and Circle Connection
Solving Trig Equations
Negative Cosine
Special Trig Relationships
Exam Type Questions
created by Mr. Lafferty
Starter
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National 5
Q1.
How can we tell if two lines are parallel.
Q2. Write down the three ratios connecting
the circle , arc length and area of a sector.
created by Mr. Lafferty
Solving Trig Equations
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National 5
Learning Intention
Success Criteria
1. We are investigating the
connect between the circle
and trig functions.
created by Mr. Lafferty
1.
Understand the connection
between the circle and sine,
cosine and tan functions.
2.
Solve trig equations using
graphically.
Trig and Circle Connection
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Sin +ve
All +ve
180o - xo
1
Sine Graph
Construction
2
3
Cosine Graph
Construction
4
Tan Graph
Construction
created by Mr. Lafferty
180o + xo
360o - xo
Tan +ve
Cos +ve
Demo
Solving Trig Equations
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National 5
Learning Intention
Success Criteria
1. We are learning how to
solve trig equations of the
form
a sin xo + 1 = 0
1.
a sin xo + 1 = 0
2.
created by Mr. Lafferty
Use the balancing method
to trig equation
Realise that there are many
solutions to trig equations
depending on domain.
Solving Trig Equations
Graphically what
a sin xo + b are
= 0 we trying to
solve
National 5
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Example :
Demo
Solving the equation sin xo = 0.5 in the range 0o to 360o
sin xo = (0.5)
xo = sin-1(0.5)
xo = 30o
There is another solution
1
2
3
4
created by Mr. Lafferty
xo = 150o
(180o – 30o = 150o)
Solving Trig Equations
Graphically what
a cos xo + b are
= 0 we trying to
solve
National 5
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Example :
Demo
Solving the equation cos xo = 0.625 in the range 0o to 360o
cos xo = 0.625
xo = cos -1 0.625
xo = 51.3o
There is another solution
1
2
3
4
created by Mr. Lafferty
(360o - 53.1o = 308.7o)
Solving Trig Equations
Graphically what
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a tan xo + b are
= 0 we trying to
solve
Example :
Demo
Solving the equation tan xo – 2 = 0 in the range 0o to 360o
tan xo = 2
xo = tan -1(2)
xo = 63.4o
There is another solution
1
2
3
4
created by Mr. Lafferty
x = 180o + 63.4o = 243.4o
Solving Trig Equations
Graphically what
a sin xo + b are
= 0 we trying to
solve
National 5
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Example :
Demo
Solving the equation 3sin xo + 1 = 0 in the range 0o to 360o
sin xo = -1/3
Calculate first Quad value
xo = 19.5o
x = 180o + 19.5o = 199.5o
There is another solution
1
2
3
4
created by Mr. Lafferty
( 360o - 19.5o = 340.5o)
Solving Trig Equations
Graphically what
National 5
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Example :
a sin xo + b are
= 0 we trying to
solve
Demo
Solving the equation 2sin xo + 1 = 0 in the range 0o to 720o
sin xo = -1/2
Calculate first Quad value
xo = 30o
xo = 210o and 330o
There are further solutions at
360o + 210o = 570o
360o + 330o = 690o
Solving Trig Equations
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National 5
Now try N5 TJ
Ex20.1 Q4 to Q10
Ch 20 (Page 198)
created by Mr. Lafferty
o
Solving Trig Equations
90
Graphically what
Higher
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Example
to
Outcome 3are we trying
A
S
o
180
solve
T C
o
Solving the equation cos2x = 1 in the range 0o to 360o 270
cos2 xo = 1
cos xo = ± 1
cos xo = 1
xo = 0o and 360o
cos xo = -1
created by Mr. Lafferty
xo = 180o
0o
Solving Trig Equations
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National 5
Now try N5 TJ
Ex20.1 Q4 onwards
Ch 20 (Page 198)
created by Mr. Lafferty
Starter Questions
Nat 5
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1. If lines have the same gradient
What is special about them.
2. Factorise x2 + 4x - 12
3. Find the missing angles.
16-Jul-15
Created by Mr. Lafferty Maths Dept.
Cosine Rule
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Nat 5
Learning Intention
1. We are learning what a
negative cosine ratio
means with respect to the
angle.
16-Jul-15
Success Criteria
1. Know what a negative cosine
ratio means.
2. Solve REAL LIFE problems
that involve finding an angle
of a triangle.
Created by Mr. Lafferty Maths Dept.
Cosine Rule
Works for any Triangle
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The Cosine Rule can be used with ANY triangle
as long as we have been given enough information.
a =b +c - 2bc cos A
2
2
2
B
a
c
A
16-Jul-15
Created by Mr Lafferty Maths Dept
C
b
Cosine Rule
Works for any Triangle
Nat 5
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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.
If YES to any of the questions then Cosine Rule
Otherwise use the Sine Rule
16-Jul-15
Created by Mr Lafferty Maths Dept
Finding Angles
Using The Cosine Rule
Works for any Triangle
Nat 5
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Consider the Cosine Rule again:
a2 = b2 +
c2 -2bc cosAo
We are going to change the subject of the formula to cos Ao
b2 + c2 – 2bc cos Ao = a2
Turn the formula around:
-2bc cos Ao = a2 – b2 – c2
Take b2 and c2 across.
2
2
2
a

b

c
cos Ao 
2bc
b c a
cos A 
2bc
2
o
2
2
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.
Finding Angles
Using The Cosine Rule
Works for any Triangle
Nat 5
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Example : Calculate the
8cm
f
unknown angle Fo .
2
2
2
d

e

f
cos F o 
2de
Fo = ?
d = 12
e = 10 f = 8
122  102  82
cos F 
2 12 10
D
E
10cm
e
12cm d
F
Write down the formula for cos Fo
Label and identify Fo and d , e and f.
o
Substitute values into the formula.
Cos Fo = 0.75
Calculate cos Fo .
Fo = 41.4o
Use cos-1 0.75 to find Fo
Finding Angles
Using The Cosine Rule
Works for any Triangle
Nat 5
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Example : Find the unknown
Angle in the triangle:
b c a
cos A 
2bc
2
2
2
C
o
Ao = yo
a = 26
b = 15
Ao =
136.3o
13cm
15cm
c
b
26cm a
B
Write down the formula.
c = 13
2
2
2
15

13

26
cos Ao 
2 15 13
cosAo = - 0.723
A
Identify the sides and angle.
Find the value of cosAo
The negative tells you
the angle is
obtuse.
Solving Trig Equations
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National 5
Now try N5 TJ
Ex20.2
Ch 20 (Page 200)
created by Mr. Lafferty
Starter
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National 5
1. Make a the subject of the formula
5 = 10b + a
2. Use the quadratic formula to solve
x2 + 5x + 1
3.
Sketch the function y = 4sin3x
created by Mr. Lafferty
Solving Trig Equations
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National 5
Learning Intention
Success Criteria
1. To explain some special
trig relationships
sin 2 xo + cos 2 xo = ?
and
tan xo and sin x
cos x
created by Mr. Lafferty
1.
Know and learn the two
special trig relationships.
2.
Apply them to solve
problems.
Solving Trig Equations
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Lets investigate
sin 2xo + cos
2
xo = ?
Calculate value for x = 10, 20, 50, 250
sin 2xo + cos
sin 2xo = 1 - cos
2
xo
2
xo = 1
Learn !
cos2xo = 1 - sin2 xo
created by Mr. Lafferty
Solving Trig Equations
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National 5
Lets investigate
tan xo
and
sin xo
cos xo
Calculate value for x = 10, 20, 50, 250
tan xo
=
sin xo
cos xo
created by Mr. Lafferty
Learn !
Given that sin
xo
3
= . Find cos xo .
5
cos2xo = 1 - sin2xo
(
cos2xo = 1 - 9
(
cos2xo = 1 - 3
5
25
cos2xo = 16
25
√
cosxo = 4
5
2
Given that cos
xo
6
=
. Find sin xo and tanxo.
10
sin2xo = 1 - cos2xo
2
6
sin2xo = 1 10
(
(
o
sin
x
tan xo =
cos xo
=
sin2xo = 1 - 36
100
sin2xo = 64
100
√
sinxo = 8
10
tan
xo
8
10
6
10
=
8
=
4
6
3
LHS
= sinxsin2x + sinxcos2x
= sinx(sin2x + sinxcos2x)
(sin2x + cos2x) = 1
= sinx = RHS
=
=
=
sinx
cosx
cosx
sinx
sinx
sinx
1
LHS
1 – sin2A = cos2A
=
=
sin2A
cos2A
tan2A = RHS
Solving Trig Equations
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National 5
Now try N5 TJ
Ex20.3
Ch 20 (Page 201)
created by Mr. Lafferty
Are you on Target !
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Higher
•
Update you log book
•
Make sure you complete and correct
ALL of the Trigonometry questions
in the past paper booklet.