p33_068.pdf

68. We use nT/ 2 to represent the integer number of half-periods speciﬁed in the problem. Note that

T =2 π/ω . We use the calculus-based deﬁnition of an average of a function:

sin 2 ( ωt

−

φ ) avg

nT/ 2

sin 2 ( ωt

−

φ ) dt

−

cos(2 ωt

−

2 φ )

2 −

2 φ )

4 ω sin(2 ωt

−

sin( nωT

2 φ ) + sin2 φ .

2 −

2 nTω

−

Since nωT = nω (2 π/ω )=2 nπ ,wehavesin( nωT

−

2 φ ) = sin(2 nπ

−

2 φ )=

−

sin2 φ so [sin( nωT

−

2 φ )+

sin2 φ ]=0. Thus,

sin 2 ( ωt

φ ) avg = 1

−

2 .