Transcript Section 5.4

Chapter 5
Analytic
Trigonometry
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SECTION 5.4 Double-Angle and Half-Angle Formulas
OBJECTIVES
1
2
3
4
Use double-angle formulas.
Use power-reducing formulas.
Use half-angle formulas.
Solve trigonometric equations involving
multiple angles and half-angles.
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DOUBLE-ANGLE FORMULAS
sin 2x  2sin x cos x
cos 2x  cos2 x  sin 2 x
cos 2x  1 2sin x
2
2 tan x
tan 2x 
2
1  tan x
cos 2x  2 cos2 x  1
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EXAMPLE 1
Using Double-Angle Formulas
3
If cos    and  is in quadrant II, find the
5
exact value of each expression.
a. sin 2
b. cos2
c. tan 2
Solution
Use identities to find sin θ and tan θ.
9 4 θ is in QII so
2
sin   1  cos   1 

25 5 sin > 0.
sin 
4/5
4
tan  


cos 3 / 5
3
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EXAMPLE 1
Using Double-Angle Formulas
Solution continued
a. sin 2  2sin cos
b. cos 2  cos   sin 
 4  3 
 2   
 5  5 
24

25
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2
2
 3  4
    
 5 5
9 16


25 25
7

25
2
5
EXAMPLE 1
Using Double-Angle Formulas
Solution continued
2 tan 
c. tan 2 
1  tan 2 
8

3

16
1
9
8

 3
7

9
 4
2  
3


2
 4
1   
 3
 8  9
    
 3  7 
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
7
6
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EXAMPLE 3
Finding a Triple-Angle Formula for Sine
Verify the identity sin 3x = 3 sin x – 4 sin3 x.
We choose the
sine-squared form
since we’re looking
for a sine-cubed
term.
Solution
sin 3x = sin (2x + x)
= sin 2x cos x + cos 2x sin x
= (2 sin x cos x) cos x + (1 – 2 sin2 x) sin x
= 2 sin x cos2 x + sin x – 2 sin3 x
= 2 sin x (1 – sin2 x) + sin x – 2 sin3 x
= 2 sin x – 2 sin3 x + sin x – 2 sin3 x
= 3 sin x – 4 sin3 x
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Omit: POWER REDUCING FORMULAS
1  cos 2x
sin x 
2
2
1  cos 2x
cos x 
2
2
1  cos 2x
tan x 
1  cos 2x
2
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Examples
4 and 5
omitted
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HALF-ANGLE FORMULAS

1  cos
sin  
2
2

1  cos
cos  
2
2

1  cos
tan  
2
1  cos
The sign, + or –, depends on the quadrant in

which lies. (Know this detail about sign.)
2
You will be provided with these
three half-angle formulas.
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EXAMPLE 6
Using Half-Angle Formulas
Use a half-angle formula to find the exact
value of cos 157.5º.
Solution
315º
Because 157.5º 
, use half-angle formula for
2


cos with   315º . 157.5º is in QII, cos  0.
2
2
Important
315º
cos157.5º  cos
2
1  cos315º

2
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EXAMPLE 6
Using Half-Angle Formulas
Solution continued
1  cos 45º

2

2  2 
22

2
1 
2
2 


22
2 2

2
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Omit Example 8.
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EXAMPLE 9
Solving a Trigonometric Equation Involving
Multiple Angles
Find all solutions in the interval [0, 2π) of the
equation 2sin  sin 3  0.
Solution
2sin  sin 3  0
3
2sin    3sin   4sin    0
2sin   3sin   4 sin   0
4 sin 3   sin   0
3
sin   4sin 2   1  0
sin   2sin   1 2sin   1  0
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EXAMPLE 9
Solving a Trigonometric Equation Involving
Multiple Angles
Solution continued
sin  0 or 2sin 1  0 or
2sin 1  0
1
1
sin   
sin  0
sin  
2
2
 7

   =
or

or
6 6
6
  0 or
 5   2    11





 
6
6
6
6
  5 7 11 
Solution set is 0, ,
, ,
.
6 6 
 6 6
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EXAMPLE 10
Solving a Trigonometric Equation Involving
Half-Angles
Find all solutions in the interval [0, 2π) of the
x
equation sin x  cos  0.
2
Solution
For x to be a solution in the interval [0, 2π), the
x
value of must be in the interval [0, π).
2
x
sin x  cos  0.
2
x
x
x
2sin cos  cos  0
2
2
2
x
x 
cos  2 sin  1  0
2
2 
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EXAMPLE 10
Solving a Trigonometric Equation Involving
Half-Angles
Solution continued
x
cos  0
2
x
2 sin  1  0
or
2
x 1
x 
sin 

2 2
2 2
x 
x 5

or

x
2 6
2
6

5
x
or x 
3
3
5 

Solution set is  ,  ,  .
3 
3
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When using
a graphing utility, restrict the
domain to the interval of interest, as in
the example using this problem . .
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We are omitting Section 5.5 Product-to-Sum
and Sum-to-Product Formulas.
Beside the
omitted power reduction
formulas and the Sum-and-Difference
and Half-Angle Formulas that you will be
given, you are to commit to memory the remaining
formulas on page 386.
You are not responsible for knowing the
eight formulas listed at the top of page
387 (from section 5.5).
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the text on the left,
These are the only
formulas you will be given
for Test I (and the FE for
this portion of the
material). They are placed
together on the
final page.
You must know or be able
to derive any of the others
you may need.
Recall that we had several
from Chapter
4, also.
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