Transcript Chapter 11: Trigonometric Identities

Chapter 11: Trigonometric Identities
11.1 Trigonometric Identities
11.2 Addition and Subtraction Formulas
11.3 Double-Angle, Half-Angle, and Product-Sum
Formulas
11.4 Inverse Trigonometric Functions
11.5 Trigonometric Equations
11.2 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.
Let 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.
11.2 Sum and Difference Identities
• By the distance formula, chords PR = SQ,
cos( A  B)  1  sin( A  B)  0
2
2
 (cos A  cos B)  (sin A  sin B) .
2
2
cos 2 ( A  B)  2 cos( A  B)  1  sin 2 ( A  B)  (cos A  cos B) 2  (sin A  sin B) 2
 2 cos(A  B)  2  cos2 A  2 cos A cos B  cos2 B  sin 2 A  2 sin Asin B  sin 2 B
 2 cos(A  B)  2  2  2 cos A cos B  2 sin A sin B
cos(A  B)  cos A cos B  sin A sin B
Simplifying this equation and using the identity
sin² x + cos² x =1, we rewrote the equation as
cos(A – B) = cos A cos B + sin A sin B.
11.2 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
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
11.2 Finding Exact Cosine Values
Example
Find the exact value of the following.
(a) cos 15°
cos15  cos(45  30)
 cos 45 cos 30  sin 45 sin 30
2 3
2 1
2 6



 
2 2
2 2
4
5 
 2 3 

cos




 12 
 12 12 
(b) cos
3 2 1 2


 
2 2 2 2
  
 cos  
6 4
 cos

6
cos

4
 sin

6
sin

4
6 2

4
11.2 Sine of a Sum or Difference
Using the cofunction relationship and letting  = A + B, we
can obtain the:
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
11.2 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
11.2 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
(a) sin 75
 sin(45  30 )




 sin 45 cos30  cos45 sin 30
2 3
2 1
2 6



 
2 2
2 2
4


11.2 Example Using Sine and Tangent Sum or Difference
Formulas
(b) tan 7  tan    
3 1

12
3
4


1  3 1


tan  tan
 2  3
3
4



1  tan tan
3
4
(c) sin 40° cos 160° – cos 40° sin 160°
= sin(40° – 160°)
= sin(–120°)
3

2
11.2 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  cos2 A  1
16
 cos 2 A  1
25
16
cos A   1 
25
3 Since cos A < 0
cos A   in Quadrant II.
5
So, sin( A  B)
 sin A cos B  cos A sin B
4  5   3  12 
         
5  13   5  13 
16

65
11.2 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 .
tan A  tan B
tan(A  B) 
1  tan A tan B
4 12
 
16
3
5


63
 4  12 
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.
11.2 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  .05.
(c) For what value of  will the graph of V = 163cos(t – ) be the
same as the graph of V = 163 sin t?
11.2 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
2
2