Trigonometry Dr J Frost jfrosttiffinkingstonschuk Last modified 18 th April 2016 Objectives from the specification Sin Graph What does it look like 90 180 270 360 90 180 ID: 623953
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Slide1
IGCSE FM Trigonometry
Dr J Frost (jfrost@tiffin.kingston.sch.uk)
Last modified:
18
th April 2016
Objectives: (from the specification)Slide2
Sin Graph
What does it look like?
90
180
270
360
-90
-180
-270
-360
?Slide3
Sin Graph
What do the following graphs look like?
90
180
270
360
-90
-180
-270
-360
Suppose we know that
sin(30) = 0.5
. By thinking about symmetry in the graph, how could we work out:
sin(150) = 0.5
sin(-30) = -0.5
sin(210) = -0.5
?
?
?Slide4
Cos Graph
What do the following graphs look like?
90
180
270
360
-90
-180
-270
-360
?Slide5
Cos Graph
What does it look like?
90
180
270
360
-90
-180
-270
-360
Suppose we know that
cos
(60) = 0.5
. By thinking about symmetry in the graph, how could we work out:
cos
(120) = -0.5
cos
(-60) = 0.5
cos
(240) = -0.5
?
?
?Slide6
Tan Graph
What does it look like?
90
180
270
360
-90
-180
-270
-360
?Slide7
Tan Graph
What does it look like?
90
180
270
360
-90
-180
-270
-360
Suppose we know that
tan(30) = 1/
√
3
. By thinking about symmetry in the graph, how could we work out:
tan(-30) = -1/√3
tan(150) = -1/√3
?
?Slide8
Solving Trig Equations
90
180
270
360
-90
-180
-270
-360
Solve
in the range
?
0.6
?
Angle Law #1:
Slide9
Solving Trig Equations
90
180
270
360
-90
-180
-270
-360
Solve
in the range
?
?
Angle Law #2:
?Slide10
Solving Trig Equations
90
180
270
360
-90
-180
-270
-360
Solve
in the range
?
-0.3
?
Angle Law #3:
Sin and cos repeat every
Slide11
Laws of Trigonometric Functions
and
repeat every
repeats every
?
?
?
?
!Slide12
Set 4 Paper 2 Q14
Test Your Understanding
?
Solve
in the range
?
Solve
in the range
?Slide13
Exercise 1
1
Solve the following in the range
Solve the following in the range
2
a
b
c
d
e
f
g
a
b
c
d
e
f
?
?
?
?
?
?
?
?
?
?
?
?
?Slide14
Trigonometric Identities
1
Then
1
2
Pythagoras gives you...
?
?
?
Using basic trigonometry to find these two missing sides…
These two identities are all you will need for IGCSE FM.
is a shorthand for
. It does NOT mean the sin is being squared – this does not make sense as sin is not a quantity that we can square!
?Slide15
Application #1
: Solving Harder Trig Equations
Solve
in the range
The problem here is that we have two different trig functions. Is there anything we could divide by to get just one trig function?
repeat every 360
repeats every 180
?
?
?
Bro Tip:
In general, when you have a mixture of sin and cos, divide everything by cos.Slide16
Solve
in the range
Test Your Understanding
?
Solve
in the range
?
repeat every 360
repeats every 180
Slide17
Application #1
: Solving Harder Trig Equations
Solve
in the range
This looks a bit like a quadratic. What would be our usual strategy to solve!
repeat every 360
repeats every 180
?
?
?
June 2013 Paper 2 Q22Slide18
More Examples
Solve
in the range
?
Solve
in the range
?Slide19
Test Your Understanding
Solve
in the range
?
Expand and simplify
. Hence or otherwise, solve
for
?Slide20
Exercise 2
Solve the following in the range
Solve the following by first factorising
.
Solve the following:
By factorising these ‘quadratics’, solve in the range
1
2
?
?
?
?
?
?
?
?
?
?
?
3
4
N
a
b
a
b
c
a
b
c
a
bSlide21
Review of what we’ve done so far
partlySlide22
Application of identities #2
: Proofs
Prove that
repeat every 360
repeats every 180
Recall that
means ‘equivalent to’, and just means the LHS is
always
equal to the RHS for all values of
.
We want to use these…
?
?
?
?Slide23
Another Example
Prove that
June 2012 Paper 1 Q16
Bro Tip
: Whenever you have a fraction in a proof question, always add the fractions.
?
?
?
?Slide24
Test Your Understanding
Prove that
AQA Worksheet
Prove that
?
?Slide25
Exercise 3
Simplify
Write out the following in terms of
:
Prove the following:
?
?
?
?
1
2
3Slide26
sin/cos/tan of
You will frequently encounter angles of
in geometric problems. Why?
We see these angles in equilateral triangles and right-angled isosceles triangles.
?
You need to be able to calculate these in non-calculator exams.
All you need to remember:
!
Draw half a unit square and half an equilateral triangle of side 2.
For
just think about the graphs of trig functions:
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?Slide27
Example Exam Questions
? Mark SchemeSlide28
Using triangles to change between sin/cos/tan
Given that
and that
is acute, find the exact value of:
?
?
Represent as a triangle
?
?
Given that
and that
is acute, find the value of:
Test Your Understanding
?
?
Given that
and that
is acute, find the value of:
?
?
1
2