Trigonometry

1)sin 8x + cos 8x= 1 * sin 8x + 1 * cos 8x= √(1² + 1²) sin (8x + φ) , where sin φ = 1 / √(1² + 1²) = 1/√2 , φ = π/4 or 3π/4
and cos φ = 1 / √(1² + 1²) = 1/√2 , φ = π/4 or 7π/8
so φ = π/4 only , = √2 sin (8x + π/4)
2)3 sin(5πx) + 3√3 cos(5πx)= 3 [ sin(5πx) + √3 cos(5πx) ]= 3 [ √(1² + √3²) sin (5πx + φ) , where sin φ = √3 / √(1² + √3²) = √3 /2 , φ = π/3 or 2π/3
and
cos φ = 1 / √(1² + √3²) = 1/2 , φ = π/3 or 5π/3
so φ = π/3 only , = 3 [ √(1² + √3²) sin (5πx + π/3) = 6 sin (5πx + π/3)

2011-09-25 18:59:25 補充：
Typing error :

line6 : 7π/8 should be 7π/4