C H A P T E R 5 Analytic Trigonometry

488 Chapter 5 Analytic Trigonometry
96. (a) To prove the identity for sinu v we first need to prove the identityfor cosu v. Assume
0 < v < u < 2 and locate u, v, and u v on the unit circle.
C
1
1
y
u − v
D
− 1
u
O
v A
1
x
B
The coordinates of the points on the circle are:
A 1, 0, B cos v, sin v, C cosu v, sinu v,
and D cos u, sin u.
Since DOB COA, chords AC and BD are equal. By the distance formula we have:
cosu v 1 2 sinu v 0 2 cos u cos v 2 sin u sin v 2
cos 2 u v 2 cosu v 1 sin 2 u v cos 2 u 2 cos u cos v cos 2 v sin 2 u 2 sin u sin v sin 2 v
cos 2 u v sin 2 u v 1 2 cosu v cos 2 u sin 2 u cos 2 v sin 2 v 2 cos u cos v 2 sin u sin v
Now, to prove the identity for sinu v, use cofunction identities.
sinu v cos
2
cos
2
u v cos u v
2
u cos v sin u sin v
2
sin u cos v cos u sin v
2 2 cosu v 2 2 cos u cos v 2 sin u sin v
2 cosu v 2cos u cos v sin u sin v
cosu v cos u cos v sin u sin v
(b) First, prove using the figure containing points
on the unit circle.
Since chords AB and CD are each subtended by angle their lengths are equal. Equating
we have
Simplifying and solving for we have
Using sin cos we have
cosu v 1
cosu v, cosu v cos u cos v sin u sin v.
2 2 sin2u v cos u cos v2 sin u sin v2 dA, B .
2 dC, D2 cosu v cos u cos v sin u sin v
A1, 0
Bcosu v, sinu v
Ccos v, sin v
−1
Dcos u, sin u
u v,
sinu v cos
2
cos
2
u v cos u v
2
u cosv sin u sinv
2
sin u cos v cos u sin v.
1
−1
y
D
u
v
u − v
u − v
C
B
A
1
x