Download Document

7–5
Double Angle and Half Angle Identities
Fundamental Identities
Sum and Difference Identities
sin2 θ + cos 2 θ = 1
sin(−θ ) = − sinθ
cos(A + B) = cos A cosB − sin A sinB
tan2 θ + 1 = sec 2 θ
cos(−θ ) = cos θ
cos(A − B) = cos A cosB + sin A sinB
1 + cot2 θ = csc 2 θ
tan(−θ) = −tan θ
sin(A + B) = sin A cos B + sin B cos A
sin(A − B) = sin A cosB − sin B cos A
Double Angle Identities
Half Angle Identities
sin(2A) = 2sin A cos A
⎛ A⎞
1+ cos A
cos⎜ ⎟ =
⎝2⎠
2
cos(2A) = cos 2 A − sin 2 A
⎛ A⎞
1− cos A
sin⎜ ⎟ =
⎝ 2⎠
2
cos(2A) = 1− 2 sin 2 A 0
cos(2A) = 2cos 2 A − 1
tan(2 A) =
⎛ A⎞
tan⎜ ⎟ =
⎝2⎠
2tan A
1− tan2 A
1− cos A
1+ cosA
There are two ways to view each identity
If you view the identity as shown in the list with sin(2A) written first, sin(2A) = 2sin A cos A , the left side
of the identity can be replaced by the right side. The function in terms of 2A sin(2A) is replaced with a
function in terms of A, . The original value of 2A is cut in half.
the original value is cut in half
sin (2x) = 2 sin (x) cos (x)
If you reverse the order of the identity with 2sin A cos A written first, 2sin A cos A = sin(2A) , the left
side of the identity can be replaced by the right side. The function in terms of A 2sin A cos A is
replaced with a function in terms of 2A, sin(2A) . The original values of A doubled.
the original values are doubled
2 sin (x) cos (x) = sin (2x)
The second interpretation is the one most commonly used in calculus and for this reason we call the
identity the double angle formula. The examples that follow will show how each identity is used.
7 - 5 Double Angle Identities
Page 1 of 5
© 2015 Eitel
Applying the Double Angle Identities as Angle Reduction Identities
The first examples of the use of the double angle identities will be to express angle measures in
terms Ax in terms of 1x where A is a whole number
Example 1
Express sin(4 x) as a function of x
sin(2A) = 2 • sin( A ) • cos( A )
sin(4 x ) = 2• [sin(2x )] • [cos(2x )]
[
sin(4 x ) = 2• [2sin x cos x ] • 1− sin2 x
[
sin(4 A) = 4 • sin x cos x • 1− sin 2 x
]
]
Example 2
Express sin(3x) as a function of x
sin(3x ) = sin(2x + x )
use 3x = 2x +1x
sin(3x ) = [sin(2x )] • cos( x ) + sin( x ) • [cos(2x )]
use angle addition identity
[
sin(3x ) = [2sin( x ) cos( x )] • cos( x ) + sin( x ) • cos 2 x − sin2 x
]
use double angle identity
for sin(2x) and cos(2x)
sin(3x ) = 2sin x cos2 x + sinx cos 2 x − sin3 x
(
)
(
distrubute
)
sin(3x ) = 2sin x 1− sin2 x + sin x 1− sin 2 x − sin3 x
use cos2 x = 1− sin 2 x
sin(3x ) = 2sin x − 2sin3 x + sin x − sin 3 x − sin 3 x
distrubute
sin(3x ) = 2sin x − 2sin3 x + sin x − sin 3 x − sin 3 x
add like terms
sin(3x ) = 3sin x − 4 sin3 x
7 - 5 Double Angle Identities
Page 2 of 5
© 2015 Eitel
Using the Double Angle Identities to prove Identities
The derivatives in calculus are easier to compute if expressions with exponents and rational
expressions are eliminated. The examples that follow show the use of the double angle identities to
take expressions with exponents and rational expressions in terms of 1x and simplify them to an
expression in terms of 2x
Use the identities above prove each Identity.
Example 1
(sin x + cos x ) 2 = 1+ sin(2x )
FOIL
sin2 x + 2 sin x cos x + cos2 x = 1 + sin(2x )
use sin2 x + cos2 x = 1
1 + 2 sin x cos x = 1+ sin(2x )
use 2sin x cos x = sin(2x)
1 + sin(2x ) = 1 + sin(2x) + 1
Example 2
2tan x
= sin(2x)
1 + tan2 x
2tan x
sec 2 x
= sin(2x)
2sin x
cosx
1
= sin(2x)
use 1+ tan2 x = sec 2 x
convert to sin x and cosx
invert and multiply
cos 2 x
2sin x cos2 x
•
= sin(2x) cancel
cosx
1
2sin x cos x = sin(2x)
use 2sin x cos x = sin(2x)
sin(2x) = sin(2x)
7 - 5 Double Angle Identities
Page 3 of 5
© 2015 Eitel
Example 3
cot x sin2x = 2cos2 x = 2cos2 x
cos x
• 2sin x cos x = 2cos2 x
sin x
use sin2x = 2sin x cos x and cot x = cos x / sin x
cancel
2cos2 x = 2cos 2 x
Example 4
(2cos2 x − 1)
2
cos4
x
(cos
x − sin4
use 2cos 2 x − 1= cos(2x) and factor the denom.
= cos2x
(cos2x) 2
2
2
)(
2
2
x + sin x cos x − sin x
(cos2x )2
cos2x
= cos2x
cos2x = cos2x
7 - 5 Double Angle Identities
)
= cos2x
usecos2 x + sin2 x = 1 and cos2 x − sin2 x = cos(2x)
cancel
cancel
Page 4 of 5
© 2015 Eitel
Example 5
3
3
sin x + cos x
= 1− sin x cos x
sin x + cos x
(
factor: think x 3 + y 3
(sin x + cos x ) sin 2 x − 2sinx cos x + cos2 x
sin x + cos x
)
= 1− sin x cos x
sin2 x − 2sin x cos x + cos2 x = 1− sin x cos x
cancel
use sin 2 x + cos 2 x = 1
1− sin x cos x = 1− sin x cos x
Example 6
2cos2x
= cot x − tan x
use cos2x = cos2 x − sin2
sin2x
(
2 cos2 x − sin 2 x
2sin x cosx
)
distruibute
2cos2 x − 2sin 2 x
= cot x − tan x
2sin x cos x
break the fraction into 2 seperate fractions with a CD
2cos2 x
2sin 2 x
−
= − cot x − tan x
2sin x cos x
2sin x cos x
cancel
cos x
sin x
−
= cot x − tan x
sin x
cos x
cot x − tan x = cot x − tan x
7 - 5 Double Angle Identities
Page 5 of 5
© 2015 Eitel