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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