Transcript B - mor media international

Trigonometry—Law of Sines
Law of Sines: sin A
a
Proof:
sin B
b
Page 24
sin C
c
(derived from the new area formula)
B
c
a
C
Problems:
1.
60°
6

2.
x
8

x
3.
60°
7
A
b
x

45°
y
55°
9

40°
y
Trigonometry—Radian Measure
Page 25
Radians—The Other Measurement for Angles
How big is 1 radian?
A (central) angle sustains a measure of 1 radian if the length of
the intercepted arc is exactly the same as the length of the radius.
6

What is the measure of angle  in radians?
Ans: ________
s
How many radians do you think angle  is?
12
Ans: ______

r
12

6
central radius
angle
What is the relationship between , r and s?
6

arc
4
_______
2

8
_______
Find the indicated variable:
s
2 rad.
4
Ans: _____
9
1.8 rad.
r
______
20

8
______
Trigonometry—Radians and Degree Conversions
Page 26
What is the relationship between radians and degrees?
Q1: How many degrees are there in a circle?
A: 360 degrees
Q2: How many radians are there in a circle?
A: We know that  = s/r. If we go around
the whole circle, s, the arc, becomes the
circumference, which is denoted by __.

r

r
So,
Conclusion:
In degrees, there are ____ in a circle; in radians, there are ____ radians in a circle.
Therefore, ___ = ___ rad. Divide both sides by 2, we obtain ___ = ___ rad.
Trigonometry—Radians and Degree Conversions (cont’d)
Page 27
 rad. = 180°
1 rad. = ( )°
( ) rad. = 1°

1 rad. ____°

1° ____ rad.
Common angles in
radians and degrees:
In general,
a) How to convert radians to degrees: Multiply by ____
 = 180°
π
2
π
3
π
4
π
6
= 90°
= 60°
= 45°
= 30°
b) How to convert degrees to radians: Multiply by ____
Trigonometry—Area of a Sector and a Segment
Area of a Sector
Page 28
Area of a Sector—Formulas
If  is in degrees:
Area = ?
60°
O
4
Area = ?
/4
O
6
Area of a Segment

O
A
r
If  is in radians:
Area of a Segment—Formulas
Area = ?
If  is in degrees:
Area = ?
A
/4
60°
O
4
O

6
O
r
If  is in radians:
Trigonometry—Negative-Angle Identities
r

–
r
y
sin  =
cos  =
tan  =
x
–y sin (–) =
cos (–) =
tan (–) =
Conclusion:
sin (–) =
cos (–) =
tan (–) =
csc (–) =
sec (–) =
cot (–) =
Page 29
Trigonometry—Addition and Subtraction Identities
Addition Identities
cos ( + ) = cos  cos  – sin  sin 
sin ( + ) = sin  cos  + cos  sin 
tan ( + ) =
tan   tan 
1  tan  tan 
Page 30
Subtraction Identities
cos ( – ) = cos  cos  + sin  sin 
sin ( – ) = sin  cos  – cos  sin 
tan   tan 
tan ( – ) = 1  tan  tan 
Michael Sullivan, the author of the textbook, used a full page (page 409) to prove that
cos ( + ) = cos  cos  – sin  sin , which I am not going to do the proof here.(1)
What I am going to do is to verify the identity (or formula) is true if I use  = 30 and
 = 60. That is,
?
cos (30 + 60) = cos 30 cos 60 – sin 30 sin 60
The real application:
Find the value of the following without using a calculator:
1. cos 37 cos 53 – sin 37 sin 53 =
2. sin 94 cos 49 – cos 94 sin 49 =
3. sin 88 cos 62 + cos 88 sin 62 =
4.
tan 128  tan( 83)

1  tan 128  tan( 83)
Note:
1. The real reason I am not doing the proof is because it’s long and tedious (and worst of all, you probably won’t get it anyway).
Trigonometry—Proving Addition and Subtraction Identities
Addition Identities
cos ( + ) = cos  cos  – sin  sin 
sin ( + ) = sin  cos  + cos  sin 
tan ( + ) =
tan   tan 
1  tan  tan 
Page 31
Subtraction Identities
cos ( – ) = cos  cos  + sin  sin 
sin ( – ) = sin  cos  – cos  sin 
tan   tan 
tan ( – ) = 1  tan  tan 
I am (still) not going to prove that
cos ( + ) = cos  cos  – sin  sin 
but I am going prove some of the other ones here, based on the fact that we are going to
take the above identity for granted (i.e., accepting it to be true without knowing the proof).
We will also need some of our already-proven identities, namely, the cofunction identities
and the negative-angle identities:
From the Subtraction Identities:
cos ( – ) = cos  cos  + sin  sin 
Proof:
From the Addition Identities:
sin ( + ) = sin  cos  + cos  sin 
Proof:
Trigonometry—Applying Addition and Subtraction Identities
Page 32
If sin  = 3/5 (/2 <  < ) and cos  = –5/13 ( <  < 3/2), find
a) sin ( + )
b) cos ( – )
c) tan ( + )
For :
 sin  = 3/5
 cos  =
 tan  =
Solution:
a) sin ( + ) =
b) cos ( – ) =
c) tan ( + ) =
For :
 cos  = –5/13
 sin  =
 tan  =
Alternate Solution (approximate):
a) sin ( + ) =
b) cos ( – ) =
c) tan ( + ) =