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Copyright © 2011 Pearson Education, Inc.
Slide 9.2-1
Chapter 9: Trigonometric Identities and
Equations
9.1 Trigonometric Identities
9.2 Sum and Difference Identities
9.3 Further Identities
9.4 The Inverse Circular Functions
9.5 Trigonometric Equations and Inequalities (I)
9.6 Trigonometric Equations and Inequalities (II)
Copyright © 2011 Pearson Education, Inc.
Slide 9.2-2
9.2 Sum and Difference Identities
• Derive the identity for cos(A – B). Let angles A and
B be angles in standard position on a unit circle with
B < A and S and Q be the points on the terminal
sides of angels A and B, respectively.
Q has coordinates (cos B, sin B).
S has coordinates (cos A, sin A).
R has coordinates (cos (A – B), sin (A – B)).
Angle SOQ equals A – B.
Since SOQ = POR, chords PR and SQ
are equal.
Copyright © 2011 Pearson Education, Inc.
Slide 9.2-3
9.2 Sum and Difference Identities
• By the distance formula, chords PR = SQ,
cos( A  B)  12  sin( A  B)  02
 (cos A  cos B)  (sin A  sin B) .
2
2
Simplifying this equation and using the identity
sin² x + cos² x =1, we can rewrite the equation as
cos(A – B) = cos A cos B + sin A sin B.
Copyright © 2011 Pearson Education, Inc.
Slide 9.2-4
9.2
Sum and Difference Identities
• To find cos(A + B), rewrite A + B as A – (– B) and
use the identity for cos (A – B).
cos( A  B )  cos( A  ( B ))
 cos A cos(  B )  sin A sin(  B )
 cos A cos B  sin A( sin B )
 cos A cos B  sin A sin B
Copyright © 2011 Pearson Education, Inc.
Slide 9.2-5
9.2
Sum and Difference Identities
.
Cosine of a Sum Or Difference
cos(A – B) = cos A cos B + sin A sin B
cos(A + B) = cos A cos B – sin A sin B
Copyright © 2011 Pearson Education, Inc.
Slide 9.2-6
9.2 Finding Exact Cosine Values
Example Find the exact value of the following.
(a) cos 15°
(or 60° – 45°)
cos15  cos(45  30 )
 cos 45 cos30 sin 45 sin 30

5
(b) cos
12
2 3
2 1
6 2
 
 
2 2
2 2
4
5
2 3 
 


cos
 cos    cos  
12
 12 12 
6 4




 cos cos  sin sin
6
4
6
4
3 2 1 2
6 2


 

2 2 2 2
4
Copyright © 2011 Pearson Education, Inc.
Slide 9.2-7
9.2 Sine of a Sum or Difference
• Using the cofunction relationship and letting
 = A + B,
 




sin( A  B )  cos   ( A  B )   cos   A   B 
2


 2





 cos   A  cos B  sin   A  sin B
2

2

 sin A cos B  cos A sin B.
Copyright © 2011 Pearson Education, Inc.
Slide 9.2-8
9.2 Sine of a Sum or Difference
• Now write sin(A – B) as sin(A + (– B)) and
use the identity for sin(A + B).
( A  B)  sin  A  ( B) 
 sin A cos( B)  cos A sin( B)
 sin A cos B  cos A sin B
Copyright © 2011 Pearson Education, Inc.
Slide 9.2-9
9.2 Sine of a Sum or Difference
Sine of a Sum or Difference
sin(A + B) = sin A cos B + cos A sin B
sin(A – B) = sin A cos B – cos A sin B
Copyright © 2011 Pearson Education, Inc.
Slide 9.2-10
9.2 Tangent of a Sum or Difference
• Using the identities for sin(A + B), cos(A + B), and
tan(–B) = –tan B, we can derive the identities for the
tangent of a sum or difference.
Tangent of a Sum or Difference
tan A  tan B
tan( A  B) 
1  tan A tan B
tan A  tan B
tan( A  B) 
1  tan A tan B
Copyright © 2011 Pearson Education, Inc.
Slide 9.2-11
9.2 Example Using Sine and Tangent Sum
or Difference Formulas
Example Find the exact value of the following.
(a) sin 75°
(b) tan 7
12
(c) sin 40° cos 160° – cos 40° sin 160°
Solution
sin 75  sin(45  30 )
(a)
 sin 45 cos 30  cos 45 sin 30

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2 3
2 1
6 2


 
2 2
2 2
4
Slide 9.2-12
9.2 Example Using Sine and Tangent Sum
or Difference Formulas


tan  tan
7




3
4
 tan    
(b) tan
12
 3 4  1  tan  tan 
3
4
3 1

1  3 1
Rationalize the denominator and simplify.
 2  3
(c) sin 40° cos 160° – cos 40° sin 160° = sin(40° – 160°)
= sin(–120°)
3

2
Slide 9.2-13
Copyright © 2011 Pearson Education, Inc.
9.2 Finding Function Values and the
Quadrant of A + B
Example Suppose that A and B are angles in standard
position, with sin A  54 , 2  A   , and cos B   135 ,
  B  32 . Find each of the following.
(a) sin(A + B) (b) tan (A + B) (c) the quadrant of
A+B
Solution
(a) sin 2 A  cos 2 A  1
16
3 Since cos A < 0 in
2
 cos A  1  cos A  
25
5 Quadrant II.
4  5   3  12  16
sin( A  B)           
5  13   5  13  65
Slide 9.2-14
Copyright © 2011 Pearson Education, Inc.
9.2 Finding Function Values and the
Quadrant of A + B
(b) Use the values of sine and cosine from part (a) to
get tan A   43 and tan B  125 .
4 12
 
16
3
5
tan( A  B ) 

4  12  63

1     
 3  5 
(c) From the results of parts (a) and (b), we find that
sin(A + B) is positive and tan(A + B) is also
positive. Therefore, A + B must be in quadrant I.
Copyright © 2011 Pearson Education, Inc.
Slide 9.2-15
9.2 Applying the Cosine Difference
Identity to Voltage
Example Common household electric current is called
alternating current because the current alternates direction
within the wire. The voltage V in a typical 115-volt outlet can be
expressed by the equation V = 163 sin t, where  is the
angular velocity (in radians per second) of the rotating generator
at the electrical plant and t is time measured in seconds.
(a) It is essential for electric generators to rotate at 60 cycles
per second so household appliances and computers will
function properly. Determine  for these electric
generators.
(b) Graph V on the interval 0  t  0.05.
(c) For what value of  will the graph of V = 163cos(t – ) be
the same as the graph of V = 163 sin t?
Copyright © 2011 Pearson Education, Inc.
Slide 9.2-16
9.2 Applying the Cosine Difference
Identity to Voltage
Solution
(a) Since each cycle is 2 radians, at 60 cycles per second,
 = 60(2) = 120 radians per second.
(b) V = 163 sin t = 163 sin 120t.
Because amplitude is 163,
choose –200  V  200 for the
range.
(c) Since cos( x  ) cos(   x) sin x, choose   .
2
Copyright © 2011 Pearson Education, Inc.
2
2
Slide 9.2-17