Transcript Sec 4.2

Chapter 4
Analytic Trigonometry
Section 4.2
Addition and Subtraction Formulas
Relationships Among Different Angles
In the previous section we have seen the different relationships that the
trigonometric functions can have with each other. Now we are interested in the
relationships different angles can have with each other. Many of these come from
realizing certain cords of the unit circle must have the same length.
P1
s+t
Q1
Coordinates of Points
t
P0
-s
Q0
P0: (1,0)
Q0: (cos(-s),sin(-s))
(cos(s),-sin(s))
P1: (cos(s+t),sin(s+t))
Q1: (cos(t),sin(t))
The length of the cord from P0 to P1 is the same
as the length of the cord from Q0 to Q1.
(cos( s  t )  1) 2  (sin( s  t )  0) 2  (cos(t )  cos( s )) 2  (sin( t )  sin( s )) 2
cos 2 ( s  t )  2 cos( s  t )  1  sin 2 ( s  t )  cos 2 t  2 cos t cos s  cos 2 s  sin 2 t  2 sin t sin s  sin 2 s
 2 cos( s  t )  2  2  2 cos s cos t  2 sin s sin t
cos( s  t )  cos s cos t  sin s sin t
Sum and Difference of Angle Identities
From the previous identity we can get 5 others given below.
Formulas for cosine
Formulas for sine
cos( s  t )  cos s cos t  sin s sin t
cos( s  t )  cos s cos t  sin s sin t
sin( s  t )  sin s cos t  cos s sin t
sin( s  t )  sin s cos t  cos s sin t
Formulas for tangent
tan s  tan t
tan( s  t ) 
1  tan s tan t
and
tan s  tan t
tan( s  t ) 
1  tan s tan t
These formulas are used to find the sine, cosine and tangent of many other angles
you do not already know by writing the angle as either a sum or difference of
known angles (i.e. usually 30(/6),45(/4) or 60(/3) or combinations of them,
or angles corresponding to these reference angles) to find the trigonometric
function of different angles.
Example: Find sin 75
To do this we think about
how to get 75. We notice:
75 = 30 + 45
Apply the sum of angles
for sine.
Example: Find tan 15
To do this we think about
how to get 15. We notice:
15 = 60 - 45
Apply the sum of angles
for tangent.
sin 75  sin( 30  45 )
 sin 30 cos 45  cos 30 sin 45


1
2
2
2

3
2
2
2
2 6
4
tan(15 )  tan( 60  45 )
tan 60  tan 45

1  tan 60 tan 45
3 1

1 3

Example: Find cos 12
To do this we think about

how to get 12
. We notice:

12
46


Apply the sum of angles
for cosine. (This is really
15=45-30)

  cos4  6 
cos12
 cos 4 cos 6  sin


2
2
3
2


4
sin

6
2 1
2 2
6 2
4
These formulas can also be used to simplify expressions where one of the values
for the angle is known.
Example: Simplify: sin x  23 
sin x  23   sin x cos 23  cos x sin
 sin x 21   cos x
 
3
2
 sin x  3 cos x

2
2
3
These identities can also be used to verify other identities that involve the angles
inside the trigonometric functions.
Verify:
cos( x  y )  cos( x  y )  2 cos x cos y
cos( x  y )  cos( x  y )
 cos x cos y  sin x sin y  cos x cos y  sin x sin y
 2 cos x cos y