MEI Core 2
Trigonometry
Section 1: Trigonometric Functions and Identities
Notes and Examples
In this section you learn about the trigonometric functions and some trigonometric
identities.
These notes contain sub-sections on:
The trigonometric functions f
Core 2
Further differentiation and integration
Glossary
Arbitrary constant
When finding an indefinite integral, you should add a constant to the function,
as when differentiating this, the constant would differentiate to zero. For
x2
example: x dx = + c .
Core 2
Trigonometry
Section 1: Trigonometric functions and identities
Solutions to Exercise
1.
C
26
B
(i)
(ii)
10
A
BC 2 = AC 2 AB 2 = 26 2 10 2 = 576
BC = 24 cm
24 12
=
26 13
10
5
=
cos A =
26 13
24 12
=
tan A =
10
5
sin A =
10
5
=
26 13
24 12
=
cos C =
MEI Core 2
Further differentiation and integration
Section 2: Integration involving negative and
fractional indices
Multiple Choice Test
1)
1
x
7
dx =
1
+c
6 x6
1
(c) 6 + c
7x
(e) I dont know
(a)
x
2)
(a)
4
5
1
4
dx =
5
x4 + c
(b)
5
(c) 14 x 4 + c
(e) I
MEI Core 2
Trigonometry
Chapter Assessment
1. Find the angle and the length x in the triangle shown below.
B
52
12 cm
x cm
A
15 cm
C
[7]
2. Andrew walks 5 km on a bearing of 140, and then walks 3 km on a bearing of
025.
(i) How far is Andrew from his star
Core 2
Trigonometry
Section 1: Trigonometric Functions and Identities
Crucial points
1. You should know the exact values for certain angles
Make sure that you use the exact values of sin , cos and tan when
= 0, 30, 45, 60 or 90, not the rounded values fr
MEI Core 2
Trigonometry
Section 1: Trigonometric functions and identities
Multiple Choice Test
Do not use a calculator for this test.
1) What is the exact value of cos 120 ?
(a) 12
(b)
(c) 23
(e) I dont know
(d)
1
2
3
2
2) What is the exact value of sin 1
Errors highlighted in the chief examiners reports in recent years for this
topic.
Gradients and tangents
Inability to interpret meaning of an infinite gradient saying
asymptote without further clarification.
dy
= 4 x 3 , ignoring the
dx
fact that it is a
Core 2
Equation of a straight line:
y y1 m1 (x x1 )
If
If
If
2
dy
dx 2
d 2y
dx 2
d 2y
dx 2
(Check
Quadratic Formula:
x
b b2 4ac
2a
< 0 , the stationary point is a maximum
> 0 , the stationary point is a minimum
= 0 , the stationary point could be a maximu
1. Introduction to multivariate data
1.1 Books
Chateld, C. and A.J.Collins, Introduction to multivariate analysis. Chapman & Hall
Krzanowski, W.J. Principles of multivariate analysis. Oxford.2000
Johnson, R.A.and D.W. Wichern Applied multivariate statisti
3. Multivariate Normal Distribution
The MVN distribution is a generalization of the univariate normal distribution which has the
density function (p.d.f.)
1
f (x) = p
2
where
2
= mean of distribution,
f (x) =
there are
p=2
(2 )
)2
(x
2
2
)
1<x<1
= varianc
4. Hypothesis testing (Hotellings T2 -statistic)
Consider the test of hypothesis
H0 :
HA
=
=
0
1
6=
0
4.1 The Union-Intersection Principle
W accept the hypothesis H0 as valid if and only if
H0 (a) : aT
= aT
0
is accepted for all linear compounds a: [in so
5. Discriminant Analysis (Classication)
Given k populations (groups)
for a set of p measurement x.
1 ; :;
k;
we suppose that an individual from
j
has p.d.f. fj (x)
The purpose of discriminant analysis is to allocate an individual to one of the groups f j
MT3732: Multivariate Statistics
Class exercise 2
Invariance of Hotellings T2 test
for linear contrasts
The weight of cork deposited in the four directions N; E; S; W was measured for 28 trees.
The sample mean and covariance matrix were
2
3
50:54 N
646:187