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Jim Lamb ers Math 1B Fall Quarter 2004- Le ture 18 Notes
These notes orresp ond to Se tion 6.3 in the text.
Double-Angle and Half-Angle Identities
A numb er of useful identities an b e obtained from the sum identities that were intro du ed in the previous le ture.
Double-Angle Identities
Re all the sum identity for sine,
sin(x + y ) = sin x os y + os x sin y : (1)
If we let y = x, then we obtain the double-angle identity for sine,
sin(2x) = sin x os x + os x sin x = 2 sin x os x: (2)
Similarly, the sum identity for osine,
os (x + y ) = os x os y sin x sin y ; (3)
applied with y = x, yields the double-angle identity for osine,
os (2x) = os x os x sin x sin x = os 2 x sin^2 x: (4) Two variations of this identity an b e derived using the Pythagorean identity os 2 x + sin^2 x = 1 : (5)
If we repla e os 2 x in equation (4) with 1 sin^2 x, then we obtain
os (2x) = (1 sin^2 x) sin^2 x = 1 2 sin^2 x: (6)
Alternatively, we an repla e sin^2 x in equation (4) with 1 os 2 x and obtain
os (2x) = os 2 x (1 os 2 x) = 2 os 2 x 1 : (7) We an use the sum identity for tangent,
tan(x + y ) = tan^ x^ +^ tan^ y 1 tan x tan y
with y = x to obtain the double-angle identity for tangent,
tan(2x) = tan^ x^ +^ tan^ x 1 tan x tan x
= 2 tan^ x 1 tan^2 x
In summary, we have derived the following double-angle identities:
sin(2x) = 2 sin x os x (10) os(2x) = os 2 x sin^2 x (11) = 1 2 sin^2 x (12) = 2 os 2 x 1 (13) tan(2x) =
2 tan x 1 tan^2 x :^ (14)
Half-Angle Identities
The double-angle identities allow us to ompute the sine, osine or tangent of an angle 2 x, given the osine, sine or tangent of the angle x. We an reverse this pro ess to obtain half-angle identities, whi h allow us to evaluate these fun tions at an angle x= 2 given the values of these fun tions at the angle x. Let m = x=2. Then x = 2 m, and it follows from the double-angle formula for osine in equation (6) that os (2m) = 1 2 sin^2 m: (15)
Rearranging, we obtain
sin^2 m =
1 os (2m) 2 ;^ (16) whi h yields the half-angle identity for sine,
sin x 2
r 1 os x 2
The sign is determined by the quadrant in whi h the angle x= 2 lies. Similarly, it follows from the double-angle formula for osine in equation (7) that os (2m) = 2 os 2 m 1 : (18)
Rearranging, we obtain
os^2 m =
1 + os (2m) 2 ;^ (19) whi h yields the half-angle identity for osine,
os x 2
r 1 + os x 2
Using a similar approa h, in whi h we multiply and divide by
p 1 os x, we an obtain the identity
tan x 2
= 1 ^ os^ x sin x
In summary, we have derived the half-angle identities
sin x 2
r 1 os x 2
os x 2
r 1 + os x 2
tan x 2
r 1 os x 1 + os x
sin x 1 + os x (30) =
1 os x sin x (31)
where, in all ases, the sign is determined by the quadrant in whi h the angle x= 2 lies.
Example 1 Compute sin( 15 Æ^ ).
Solution The angle 15 Æ^ lies in Quadrant IV, where the sine fun tion is negative. Therefore, by the half-angle formula for sine, we have
sin( 15 Æ^ ) =
r 1 os ( 30 Æ^ ) 2
=
r 1 os 30 Æ 2
=
s 1
p 3 = 2 2
=
s 2
p 3 4
=
p 2
p 3 2 :^ (32)
In the se ond step, we used the fa t that os( x) = os x. 2
