Download Double Angle Formulas

Double Angle Formulas
T, 11.0: Students demonstrate an
understanding of half-angle and doubleangle formulas for sines and cosines and
can use those formulas to prove and/or
simplify other trigonometric identities.
Double Angle Formulas
Objectives
• Know the double angle
identities
• Verify more complex
trigonometric identities using
the basic trigonometric
identities, Pythagorean
identities, co-function
identities, odd/even identities,
sum & difference identities,
double angle identities, halfangle identities and justify
each step in the verification
process.
Key Words
• Degrees
• Radians
• Double-Angle Identities
Quick Check
Solve
Solve
Double Angle Identities
Using the Double Angle Formula
2
and  has its terminal side
5
in the first quadrant, find the
exact value of each function.
If sin  =
a. sin 2
To use the double-angle identity for sin 2, we must first find cos .
sin2  + cos2  = 1
2
2 + cos2  = 1
5
21
cos2  = 25
21
cos  = 5

sin  =
2
5
Then find sin 2.
sin 2 = 2 sin  cos 
2 21
2
21
= 2(5)( 5 ) sin  = 5, cos  = 5
4 21
= 25
b. cos 2
Since we know the values of cos  and sin , we can use any of the double-angle identities for
cosine.
cos 2 = 2 cos2  - 1
2


= 2  21  - 1
 5 
17
= 25
21
cos  = 5
Using the Double Angle Formula
2
If sin  = and  has its terminal side
5
in the first quadrant, find the
exact value of each function.
c. tan 2
We must find tan  to use the double-angle identity for tan 2.
sin 
tan  =
cos 
2
5
2
21
=
sin  = 5, cos  = 5
21
5
2
2 21
=
or 21
21
Then find tan 2.
2 tan 
tan 2 =
1 - tan2 


2  2 21 
2 21
 21 
=
tan  = 21
2


1   2 21 
 21 
4 21
21
4 21
= 17 or 17
21
d. sin 4
Since 4 = 2(2), use a double-angle identity for sine again.
sin 4 = sin 2(2)
= 2 sin (2) cos (2)
4 21 17
= 2( 25 )(25)
136 21
=
625
Double-angle identity
4 21
17
sin (2) = 25 , cos (2) = 25
You Try
2
If sin 𝜃 = and 𝜃 has its
3
terminal side in the first
quadrant, find the exact
value of each function.
a) sin 2𝜃
b) cos 2𝜃
c) tan 2𝜃
d) cos 4𝜃
You Try
2
If sin 𝜃 = and 𝜃 has its
3
terminal side in the first
quadrant, find the exact
value of each function.
a) sin 2𝜃
b) cos 2𝜃
c) tan 2𝜃
d) cos 4𝜃
Verify Trigonometric Identity
-1 - cos 2
Verify that
= -cot  is an identity.
sin 2
-1 - cos 2 ?
 -cot 
sin 2
-1 - (2 cos2  - 1) ?
 -cot 
2 sin  cos 
-2 cos2  ?
 -cot 
2 sin  cos 
cos  ?
 -cot 
sin 
-cot  = -cot 
Double-angle identities for cosine and sine
You Try
Conclusions
Summary
• Verify that
Sin2x cosx – cos2x sinx = sinx
is an identity.
Assignment
• Double Angle Formulas
– Page 453
– #(8-12,21-23,28-30)