3.4 Trigonometric relations 75
Also, puttingx 1 = 1 y,
sin 12 x 1 = 121 sin 1 x 1 cos 1 x (3.23)
cos 12 x 1 = 1 cos
2x 1 − 1 sin
2x 1
= 111 − 121 sin
21 x 1 = 121 cos
21 x 1 − 11
(3.24)
where the alternative expressions for cos 12 xare obtained fromsin
2x 1 + 1 cos
2x 1 = 1 1.
EXAMPLE 3.11Expresssin 15 θandsin 1 θin terms of the sines and cosines of 2 θ
and 3 θ.
From equations (3.21),
sin 15 θ 1 = 1 sin(3θ 1 + 12 θ) 1 = 1 sin 13 θ 1 cos 12 θ 1 + 1 cos 13 θ 1 sin 12 θ
sin 1 θ 1 = 1 sin(3θ 1 − 12 θ) 1 = 1 sin 13 θ 1 cos 12 θ 1 − 1 cos 13 θ 1 sin 12 θ
0 Exercises 18–21
EXAMPLE 3.12Expresssin 13 θ 1 cos 12 θin terms ofsin 1 θandsin 15 θ.
From equations (3.21) it follows that
(3.25)
and, therefore,
0 Exercises 22
EXAMPLE 3.13Expresssin 13 θ 1 sin 12 θandcos 13 θ 1 cos 12 θin terms ofcos 1 θandcos 15 θ.
From equations (3.22) it follows that
sin 1 xsin 1 y
(3.26)
and, therefore,
sin 13 θ 1 sin 12 θ
0 Exercises 23
cos cos cos cos 32
1
2
θθ=+θ θ 5
=−
1
2
cos cosθθ 5
cos cosx y=−++cos(xy xy) cos( )
1
2
=−−+cos(xy xy) cos( )
1
2
sin cos sin sin 32
1
2
θθ=+ 5 θ θ
sin cosx y=++−sin(xy xy) sin( )
1
2