Download SYLLABUS

INTERNATIONAL ISLAMIC UNIVERSITY MALAYSIA
CENTRE FOR FOUNDATION STUDIES
MATHEMATICS II
( SHF 1124 )
SYLLABUS
1
SYLLABUS
MATHEMATICS II
SHF1124
Aims
The Mathematics II syllabus aims to develop the understanding of trigonometry and
statistical concepts, together with the skills in mathematical reasoning and problem
solving, so as to enable students to pursue their studies in programmes related to
Biological Sciences at the Kulliyyah.
Course Objectives
The objectives of this syllabus are to develop the abilities of students to
1) apply mathematical terms, notations, principles, statistics applications and
mathematical methods.
2) analyze, interpret and make statistical decisions in the field of science, technology
and social management.
3) develop understanding of mathematical concepts and skills to interpret and solve
problems.
4) provide students with a strong foundation to pursue studies in Medical, Pharmacy,
Allied Health Sciences, Nursing, Dentistry and Biological Science programmes.
Content
2
1.
TRIGONOMETRIC FUNCTIONS (10 hours)
1.1 Angles and their measure (1 hour)
1.2 Trigonometric Functions: The Unit Circle Approach (2 hours)
1.3 Properties of the Trigonometric Functions (3 hours)
1.4 Trigonometric Graphs (4 hours)
Learning Outcomes:
Students should be able to
a) convert from degrees to radians and from radians to degrees;
b) define the trigonometric functions sinθ, cosθ, tanθ, cosecθ, secθ and cotθ
using terminal points determined by t on the unit circle and right triangle;
c) find and use the exact values of the trigonometric functions of quadrantal angles
and integer multiples of quadrantal angles;
d) find

4
and
 45 ,

6
use
 30 ,
the

3
exact
values
of
the
trigonometric
functions
of
 60
e) find exact values for certain integer multiples of

4
 45 ,

6
 30 ,

3
 60
f) determine the period of trigonometric functions;
g) determine the signs of trigonometric functions in a given quadrant;
h) find the exact values of trigonometric functions using reference angle,
fundamental identities, periodic properties and by using even-odd properties;
i) find exact values of the trigonometric functions of an angle given one of the
functions and the quadrant of the angle;
j) graph sine, cosine, and tangent functions;
k) determine the domain and range of the trigonometric functions;
l) determine the amplitude and period of sinusoidal functions;
m) sketch the graph of sinusoidal functions in the form of:
y = Asin(x)
y = Acos(x)
y = Asin(ωx – Φ)
y = Acos(ωx – Φ)
3
n) find an equation for a sinusoidal graph.
2.
ANALYTIC TRIGONOMETRY (15 hours)
2.1 Inverse Trigonometric Functions (4 hours)
2.2 Trigonometric Identities (2 hours)
2.3 Sum and Difference Formulas (2 hours)
2.4 Double Angle and Half-Angle Formulas (2 hours)
2.5 Product-to-Sum and Sum-to-Product Formulas (2 hours)
2.6 Trigonometric Equations (3 hours)
Learning Outcomes:
Students should be able to
a) define the inverse functions y  sin 1 x, y  cos 1 x, y  tan 1 x by explaining one-toone function and domain restriction;
b) find exact values of the inverse sine, cosine, and tangent functions, example: sin 1
1
2
;
c) find an approximate value of the inverse sine, cosine, and tangent functions using
calculator;
d) find the exact value of a composite trigonometric function and their inverse, example
5
 7 

1   7
cos  tan 1  , cos 1 sin
 , sin sin  
6 
12 


  6

 ;

e) find the exact value of expressions involving the inverse sine, cosine, and tangent

5
 3 
functions example: cos  tan 1  sin 1   ;
12
 5 

f) use calculator to evaluate sec-1x, csc-1x, cot-1x;
4
g) write a trigonometric expression as an algebraic expression;
h) use algebra to simplify trigonometric expressions;
i) establish identities by using the quotient, reciprocal, pythagorean, and even-odd
identities;
j) use sum and difference formulas to find exact values;
k) use sum and difference formulas to establish identities;
l) use sum and difference formulas involving inverse trigonometric functions;
m) use Double-Angle Formulas for sin2A, cos2A, tan2A to find exact values and to
establish identities;
n) use

2
Half-angle

Formulas
sin

2

1  cos
,
2
1  cos 
sin 
1  cos

=
sin 
1  cos 
1  cos
cos

2

1  cos 
,
2
tan
to find exact values and to establish
identities;
o) express products as sum and vice versa by using the Product-To-Sum and the SumTo-Product Formulas;
p) Use Product-To-Sum and the Sum-To-Product Formulas to establish identities;
q) solve trigonometric equations involving a single trigonometric function example sin
1
2
 = , cos(2  ) = 

1
3
, tan(3   )  ;
4
2
2
r) solve trigonometric equations quadratic in form;
s) solve trigonometric equations using identities;
t) solve trigonometric equations linear in sine and cosine acosθ + bsinθ = c by
expressing acosθ ± bsinθ as R cos(   ) or R sin(    ) ;
3.
STATISTICS (10 hours)
3.1 The Nature of Probability and Statistics (1 hour)
3.1.1 Descriptive and Inferential Statistics
3.1.2 Variables and Types of data
5
3.1.3 Data collection and Sampling techniques
3.2 Frequency Distributions and Graphs (3 hours)
3.2.1 Organizing data
3.2.2 Histograms, Frequency Polygons and Ogives
3.3 Data description (6 hours)
3.3.1 Measures of Central Tendency
3.3.1 Measures of Variation
3.3.3 Measures of Position
Learning Outcomes:
Students should be able to
a) identify the basic statistical terms;
b) differentiate between descriptive and inferential statistics;
c) distinguish between sample and population;
d) identify the nature of variables and types of data;
e) identify the measurement level for each variable and the four basic sampling
techniques;
f) organize data using frequency distributions;
g) construct the three types of frequency distributions: categorical frequency
distribution, grouped frequency distribution and ungrouped frequency distribution;
h) represent data in frequency distributions graphically using histograms, frequency
polygons and ogives;
i) construct histogram, frequency polygons and ogives by using relative frequency;
j) find values below a certain data from ogive;
k) find mean, median and mode for grouped and ungrouped data;
l) describe ungrouped data using range, variance and standard deviation;
m) identify the standard deviation and variance formula for population and sample;
n) find standard deviation and variance for grouped data;
o) identify the position of a data value in a data set, using standard scores, percentiles,
deciles and quartiles;
p) find percentile corresponding to a given value in a data set;
6
q) construct a percentile graph;
4.
PROBABILITY AND COUNTING RULES (15 hours)
4.1 Counting Rules (5 hours)
4.2 Sample spaces & Probability (2 hours)
4.3 The Addition Rules for Probability (2 hours)
4.4 The Multiplication Rule and Conditional Probability (3 hours)
4.5 Probability & Counting Rules (3 hours)
Learning Outcomes:
Students should be able to
a) find the total number of outcomes in a sequence of events, using the fundamental
counting rule;
b) find the total number of outcomes in a sequence of events using tree diagram;
c) use the factorial notation;
d) find the number of ways that r objects can be selected from n objects, using the
permutation rule;
e) find the number of ways that r objects can be selected from n objects without regard
to order, using the combination rule;
f) determine sample spaces by experiment or by using tree diagram;
g) find the probability of an equally likely event , using classical probability or empirical
probability;
h) identify the probability properties;
i) find the probability of complementary events by using formula and showing Venn
Diagram;
j) find
the
probability
of
compound
P( A  B)  P( A)  P( B)  P( A  B) ;
7
events,
using
addition
rules
k) identify the mutually exclusive events;
l) use Venn Diagram for the addition rules;
m) identify independent and dependent events;
n) find the probability of compound events, using the multiplication rules;
o) find the conditional probability of an event;
p) find the conditional probability using tree diagram and Venn Diagram;
q) solve probability problems involving the term ‘at least’;
r) find the probability of an event, using counting rules.
5.
DISCRETE PROBABILITY DISTRIBUTIONS (10 hours)
5.1 Probability distributions (2 hours)
5.2 Mean, Variance and Expectation (2 hours)
5.3 The Binomial distribution (3 hours)
5.4 The Poisson distribution (3 hours)
Learning Outcomes:
Students should be able to
a) construct a probability distribution for a random variable;
b) identify the discrete probability distribution;
c) show probability distribution graphically;
d) identify the requirements for a probability distribution;
e) find the mean, variance and expected value for a discrete random variable;
f) identify the requirements for a binomial experiment;
g) find the exact probability for X successes in n trials of a binomial experiment by
using formula and table;
h) find the mean, variance and standard deviation for the variable of a binomial
distribution;
i) identify conditions when Poisson distribution is used;
8
j) find probability using the Poisson distribution formula;
k) use the Poisson formula to approximate the binomial distribution;
6.
THE NORMAL DISTRIBUTION (10 hours)
6.1 Properties of the Normal distribution (1 hour)
6.2 The Standard Normal Distribution (2 hours)
6.3 Applications of the Normal distribution (2 hours)
6.4 Central Limit Theorem (3 hours)
6.5 The Normal Approximation to the Binomial Distribution (2 hours)
Learning Outcomes:
Students should be able to
a) identify the properties of the normal distribution;
b) find the area under the standard normal distribution, given various z values;
c) find the z value, given the area under the standard normal distribution;
d) find probabilities for a normally distributed variable by transforming it into a standard
normal variable;
e) find specific data values for given percentages, using the standard normal
distribution;
f) identify the sampling distribution of sample means;
g) explain the sampling error generally;
h) identify the properties of the distribution of sample means;
i) use the central limit theorem to solve problems involving sample means for large
samples;
j) compare the binomial distribution and a normal distribution;
k) use the normal approximation to compute probabilities for a binomial variable;
9
l) identify instances where the correction for continuity is needed.
Form of Examination
The end-of-semester examination contains 10 compulsory questions of variable mark
allocations totaling 100 marks. The duration of the paper is 3 hours.
Reference Books
1) Sullivan,M., Precalculus (Eighth Edition), Pearson, Prentice Hall, 2008.
2) Bluman, A.G., Elementary Statistics – A step by step Approach (Seventh Edition),
McGraw Hill, 2007.
3) Stewart,J., Redlin,L., Watson,S., Precalculus (Fifth Edition), Thomson Brooks/Cole,
2006.
4) Tan, C.E., Ong, B.S., Khor, B.H., Lye, M.S., Mathematics for Matriculation
Semester2, Penerbit Fajar Bakti Sdn. Bhd., 2005.
10