Engineering MATHEMATICS

MET 3403

1.Trigonometric Functions Every right-angled triangle contains two acute angles. With respect to each of these angles, there are six

functions, called trigonometric functions, each involving the lengths of two of the sides of the triangle.

Consider the following triangle ABC

AC is the side adjacent to angle a, BC is the side opposite to angle a.

Similarly, BC is the side adjacent to angle b, AC is the side opposite to angle b.

B

C A

hypotenuse

a

adjacent

opposite

Six trigonometric functions with respect to angle a:

Note:

x

y

AC

BC

adjacent

opposite

r

x

AB

AC

hypotenuse

adjacent

r

y

AB

BC

hypotenuse

opposite

)atan(

)acos(

)asin(

B

C A

Hypotenuse (r)

a

Adjacent (x)

opposite (y)

)tan(

1)cot(

)cos(

1)sec(

)sin(

1)csc(

aa

aa

aa

AB

AC

opposite

adjacent

AC

AB

adjacent

hypotenuse

BC

AB

opposite

hypotenuse

)tan()cos(

)sin(a

a

a )cot(

)sin(

)cos(a

a

a

Example: Consider the right-angled triangle, with lengths of sides indicated, find sin(d), cos(d), tan(d), sin(e), cos(e), tan(e).

5

12)tan(

13

5)cos(

13

12)sin(

DF

EFd

ED

DFd

ED

EFd

12

5)tan(

13

12)cos(

13

5)sin(

EF

DFe

ED

EFe

ED

DFe

F D

13

d

5

12

e

E

Pythagorean Theorem (畢氏定理 )

222 ABACBC

B

C A

Pythagorean Identities derived from Pythagorean theorem

)(csc1)(cot

)(sec1)(tan

1)(cos)(sin

22

22

22

aa

aa

aa

Example: In right-angled triangle, sin(a)=4/5, find the values of

the other five trigonometric functions of a.

Since sin(a)=opposite over hypotenuse=4/5

4

5)csc(,

3

5)sec(,

4

3)cot(,

3

4)tan(,

5

3)cos(

345

5,4

2222

aaaaa

BCABAC

ABBC

A

B

C

54

a

Example: If, in right-angled triangle, sin(a)=7/9, find the values

of cos(a) and tan(a). Using trigonometric identity,

Since

9

24

81

32)cos(

9

71)(cos

1)(cos)(sin2

2

22

aa

aa

8

27

24

7

924

97)tan(

)cos(

)sin()tan(

a

a

aa

Angle in degree Each degree is divided into 60 minutes Each minute is divided into 60 seconds Example:

Express the angle 265.46 in Degree-Minute-Second (DMS) notation

"36'27265'1

"60'6.0'27265

1

'6046.026546.265

Angle in radian A unit circle has a circumference of 2 One complete rotation measures 2 radian

Angle of 360 = 2 radian

Example:

45180

44

61803030

Special Angles (1) For a 30-60-90 right-angled triangle

From the triangle,

3)60tan(,2

1)60cos(,

2

3)60sin(

3

3

3

1)30tan(,

2

3)30cos(,

2

1)30sin(

A

B

C

21

30◦

60◦

3

Special Angles (2) For a 45-45-90 right-angled triangle

From the triangle,

1)45tan(,2

2

2

1)45cos()45sin(

A

B

C1

1

45◦

45◦

2

Unit circle and sine, cosine functions Start measuring angle from positive x-axis

‘+’ angle = anticlockwise

‘’ angle = clockwise θ

(x,y)(x,y)

’’ x

y

0

(x,y)

(x,y)

’’

x

y

0

(x,y)

(x,y)

’’

x

y

0

Quadrant IIQuadrant I

Quadrant IVQuadrant III

Angle and quadrants Value of a trigonometric function for an angle in 2nd,

3rd or 4th quadrants is equal to plus or minus of the value of the 1st quadrant reference angle

Quadrant II Quadrant I

Quadrant IVQuadrant III

The sign of the value is dependent upon the quadrant that the angle is in.

ALL +veSINE +ve

COSINE +veTANGENT +ve

Exercise: find WITHOUT calculator:

sin(30 °) = _________cos(45 °) = _________tan(315 °)= _________sin(60 °) = _________cos(180 °) = _________tan(135 °)= _________sin(240 °) = _________cos(-45 °) = _________

Hint

3)60tan(,2

1)60cos(,

2

3)60sin(

3

3

3

1)30tan(,

2

3)30cos(,

2

1)30sin(

•Simple trigonometric equations

Notation : If sin = k then = sin-1k (sin-1 is written as inv sin or

arcsin). Similar scheme is applied to cos and tan.

e.g. Without using a calculator, solve sin = 0.5, where 0o

360o

e.g. Solve cos 2 = 0.4 ,where 0 2