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IGCSE FM IGCSE FM

IGCSE FM - PowerPoint Presentation

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IGCSE FM - PPT Presentation

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

360 180 270 solve 180 360 solve 270 range sin cos graph trig tan repeat solving test equations prove

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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