POLITEKNIK UNGKU OMAR CONTROL GROUP

ENGINEERING MATHEMATICS 1 DBM1013

1

Trigonometry Syllabus

Matrices Syllabus

2

Define sine, cosine, tangent, secant, cosecant and cotangent

Sketch the graph of sine, cosine and tangent

Graph y = sin x Graph y = cos x

x 2

0 2

23

2

x 2

0 2

2

3

2

sin x 1 0 1 0 1 0 cos x

Graph y = tan x

x 2

0 2

23

2

sin x

sin =

cos =

tan =

csc =

sec =

cot =

A

B

C

3

Positive and Negative Values of Trigonometric Functions in Quadrant

Find the values of trigonometric functions

i. cos 120 = ________

ii. tan 220 = ________

iii. sin 320 20 = ________

iv. cos = 0.9205. Find between 3600 .

v. sin = -0.5736. Find between 270180 .

vi. tan = -1.048. Find between 3600 .

4

Solve trigonometric equations involving trigonometric basic identities, compound angle and double

angle formulae

Find the value of that will satisfy the following equations ( 3600 ).

i. 01sin2

ii. 01sinsin2 2

iii. 03sin3cos5 2

iv. 0sin2sin

Trigonometric Basic Identities

1cossin 22

22 sectan1

22 cos1cos ec

Double Angle Formulae

AAA cossin22sin

AAA 22 sincos2cos

A

AA

2tan1

tan22tan

Compound Angle Formulae

ABBABA cossincossin)sin(

ABBABA sinsincoscos)cos(

BA

BABA

tantan1

tantan)tan(

5

v. sin2cos

By using Compound Angle Formulae, expand and simplify 60cos .

Prove that xx cos90sin

Sine Rule and Cosine Rule

Find the length of x in the triangle below and figure out the area of it.

Sine Rule

C

c

B

b

A

a

sinsinsin

Cosine Rule

Abccba cos2222

60

80

7 cm

x

Area of a Triangle

AbcA sin2

1

6

Identify the character of matrices

Matrix A =

620

430

441

Elements of matrix A are the numbers 1, 4, -4, 0, -3, 4, 0, -2 and 6.

Element 13a is -4.

Matrix B =

052

763. The order of matrix B is 32 .

Types of matrices

Square Matrix is a matrix with the same number of rows and columns. Example:

4

104

24

620

430

441

Zero Matrix or null matrix is a matrix with all the elements being zero. Example:

0

00

00

000

000

000

Diagonal Matrix is a square matrix which has zeroes everywhere other than the main diagonal.

70

02

300

020

005

Identity Matrix is a square matrix which has a 1 for each element on the main diagonal and 0 for all other elements.

100

010

001

Transpose of a Matrix is a matrix which is formed by turning all the rows of a given matrix into columns and vice versa.

Matrix A =

052

763 Transpose of matrix, AT =

07

56

23

Row

Column

Row number Column number

Number

of row

Number

of column

7

Addition of Matrices

To add two or more matrices, the order of the matrices must be similar.

1031

0314

0852

3517

2331

10413

Subtraction of Matrices

1031

0314

0852

3517

2331

10413

Multiplication of Matrices

Multiplication of matrices can only be done when the number of column in matrix A is equal to the number of row in matrix

B.

A =

943

852

761

B =

43

52

61

33

23

A =

70

02 B =

07

56

23

22

23

43

52

61

943

852

761

8

Determinant of matrices

A determinant is a real number associated with every square matrix. It is denoted by “det A” or |A |

22 A =

db

ca, |A| = bcad

33

B =

ifc

heb

gda

, |B| = ecbfghcbidhfeia

Find the determinant of matrix C.

C =

052

640

241

Inverse of matrices

A-1 = A

1adj.A

Step 1: Calculating the matrix of minors

Step 2: Convert into matrix of cofactors

Step 3: Find the adjoin matrix

Step 4: Multiply by 1/determinant

A =

212

034

231

Step 1:

Minor A =

34

31

04

21

03

23

12

31

22

21

21

23

12

34

22

04

21

03

=

986

524

266

9

Step 2:

986

524

266

986

524

266

Minor A Cofactor A

Step 3:

986

524

266

952

826

646

Cofactor A Adjoint A

Step 4:

Solve simultaneous linear equations by using Inverse method

Solve the following simultaneous linear equations:

55

1123

172

zyx

zyx

zyx

Step 1: Write in this form.

5

11

17

151

123

321

z

y

x

Step 2: To find

z

y

x

, we need to find the inverse of the matrix

151

123

321

.

5

11

17

151

123

3211

z

y

x

10

Step 3: Finding the inverse.

Solve simultaneous linear equations by using Cramer’s Rule

11

Use Cramer’s rule to solve this.

55

1123

172

zyx

zyx

zyx