Transcript CLASS 1 CHARACTERISTICS of FUNCTIONS, ALGEBRAICALLY

Chapter 7 – Analytic Trigonometry
7.3 - Double-Angle, Half-Angle, and Product-Sum
Formulas
Double Angle Formulas

The double-angle formulas allow us to find the values
of the trigonometric functions at 2x from their values
at x. Formulas for Sine:
sin 2 x  2sin x cos x
Formulas for Cosine:
cos 2 x  cos 2 x  sin 2 x
 1  2sin 2 x
 2 cos 2 x  1
Formulas for Tangent:
tan 2 x 
2 tan x
1  tan 2 x
7.3 - Double-Angle, Half-Angle, and Product-Sum
Formulas
Proofs of Double Angle
Formulas
sin 2  sin    
 sin  cos   cos  sin 
 2sin  cos 
Now, you try the proof for cos2 and tan 2.
cos 2  cos    
tan 2  tan    
 cos  cos   sin  sin 
 cos2   sin 2 
7.3 - Double-Angle, Half-Angle, and Product-Sum
Formulas
tan   tan 

1  tan  tan 
2 tan 

1  tan 2 
Example
 If sin  
4
5
and  lies in quadrant II, find the exact
value of each of the following:
sin 2
cos 2
tan 2
7.3 - Double-Angle, Half-Angle, and Product-Sum
Formulas
Example

Find the exact value of
cos 15  sin 15
2
2
7.3 - Double-Angle, Half-Angle, and Product-Sum
Formulas
Power Reducing Formulas

The power reducing formulas play an important part
in calculus.
Formulas for Sine:
1  cos 2 x
sin x 
2
Formulas for Cosine:
cos 2 x 
Formulas for Tangent:
1  cos 2 x
tan x 
1  cos 2 x
2
2
7.3 - Double-Angle, Half-Angle, and Product-Sum
Formulas
1  cos 2 x
2
Half Angle Formulas
u
1  cos x

2
2
Formulas for Sine:
sin
Formulas for Cosine:
u
1  cos x
cos  
2
2
Formulas for Tangent:
u
1  cos x 1  cos x
sin u
tan  


2
1  cos x
sin u
1  cos u

NOTE: The choice of the + or – sign depends on the
quadrant in which u/2 lies.
7.3 - Double-Angle, Half-Angle, and Product-Sum
Formulas
Example

Find the exact value of cos105o.
7.3 - Double-Angle, Half-Angle, and Product-Sum
Formulas
Example
Verify the identity:
sin 2
tan  
1  cos 2
7.3 - Double-Angle, Half-Angle, and Product-Sum
Formulas
Example
Verify the identity:

sec 
tan 
2 sec  csc   csc 
7.3 - Double-Angle, Half-Angle, and Product-Sum
Formulas
Product-to-Sum Formulas
sin u cos v 
1
sin  u  v   sin  u  v  
2
1
cos u sin v  sin  u  v   sin  u  v  
2
1
cos u cos v  cos  u  v   cos  u  v  
2
1
sin u sin v  cos  u  v   cos  u  v  
2
7.3 - Double-Angle, Half-Angle, and Product-Sum
Formulas
Examples – pg. 515

Write the product as a sum.
55. sin 2 x cos 3 x
x
x
60. 11sin cos
2
4
7.3 - Double-Angle, Half-Angle, and Product-Sum
Formulas
Sum-to-Product Formulas
sin x  sin y  2sin
x y
x y
cos
2
2
x y
x y
sin x  sin y  2 cos
sin
2
2
x y
x y
cos x  cos y  2 cos
cos
2
2
x y
x y
cos x  cos y  2sin
sin
2
2
7.3 - Double-Angle, Half-Angle, and Product-Sum
Formulas
Examples – pg. 515

Write the product as a sum.
61. sin 5 x  sin 3 x
63. cos 4 x  cos 6 x
7.3 - Double-Angle, Half-Angle, and Product-Sum
Formulas
Example – pg. 515

Prove the identity.
sin x  sin y
 x y
89.
 tan 

cos x  cos y
 2 
7.3 - Double-Angle, Half-Angle, and Product-Sum
Formulas
Example

Write an equivalent expression for cos4x that does not
contain powers of trigonometric functions greater
than 1.
7.3 - Double-Angle, Half-Angle, and Product-Sum
Formulas
More Examples

Prove the identity.
74. sin 8 x  2sin 4 x cos 4 x
1  tan 2 x
80. cot 2 x 
2 tan x
sin x  sin 5 x
85.
 tan 3 x
cos x  cos 5 x
7.3 - Double-Angle, Half-Angle, and Product-Sum
Formulas