Mathematics in Education and
Industry
Warm Up…
Warm Up…
Find all 4 real values of x that satisfy
(X-5)
x2- 4
=1
AEA Session 1: Constructing a beautiful, clear,
concise argument: the proper use of notation
• how to write mathematics in “good style”
• understanding key symbols and using
them correctly
• ‘exact’ values
• reasoning, conciseness and clarity
Warm Up
Common symbols with which you should be familiar:

“implies”, “means that” – very useful for linking
statements together and can help avoid the tendency to
overuse the “equals” sign






“not equal to” – hardly ever used by students but
surprisingly useful
“identically equal to” – so signifies an identity, something
which is true for all values, as opposed to an equation
“approximately equal to”
“because”
“therefore”
“the SUM of” (“sigma”) – eg:
arcsin x
sin
1
x

 nx
n
n 1
equivalent statements for “inverse sine”, ie: “the angle
whose sine is …”
Exact values:
Fill in the table!
sin
0

6

4

3

2
cos
tan
Exact values:
Learn these (or know how to quickly derive them)
0

6

4

3

2
sin
cos
tan
0
1
0
1
2
1
2
3
2
1
3
1
3
2
1
2
1
2
1
0
3

AEA Specimen Paper Q4
8 Marks
The notation in the printed answer looks daunting – it is meant to be! But this
question relies only on standard co-ordinate geometry techniques.
AEA June 2004 Q2
2 Marks
2 Marks
This question makes use of the “sigma” notation and requires a fairly standard
application of the binomial theorem
AEA June 2007 Q3
(a)
Solve, for 0 ≤ x < 2π,
cos x  cos 2x  0
(b)
5 Marks
Find the exact value of x, x ≥ 0, for which
arccos x  arccos 2 x 

2
6 Marks
This question uses the “arc” notation, meaning “inverse cos” or “the angle whose
cosine is ..” (alternative symbol cos -1 x); you should aim to answer the first part as
concisely as possible whilst still maintaining clarity and reason. The second part is a
bit more of a challenge …