Basic Trigonometry, 基本三角

Your provided answer is not correct.
For the sake of simplicity, I use X = 4000*pi*t
Therefore the expression is sinX + cos2X
= sinX + (1 - 2sin^2X)
= - 2sin^2X + sinX + 1
= -2(sin^2X - sinX / 2 + 1/16) + 1 + 1/8
= -2(sinX - 1/4)^2 + 9/8
Obviously the maximum of the expression happens when the bracketed terms is zero, i.e. when sinX = 1/4 and the maximum value is 9/8 = 1.125
The minimum value is when sinX = -1, and the minimum value
= -2(-1 - 1/4)^2 + 9/8
= -2
Probably you need to check your question for correctness. If your question is
y = sin(4000*pi*t) - cos(8000*pi*t) then the maximum is 2.
I plot graph for the expression y = sin(4000*pi*t) + cos(8000*pi*t) and verify my calculation is correct.

圖片參考：http://img525.imageshack.us/img525/7209/sinecosine.png

find the maximum amplitude of y(t),y(t) = sin(4000*pi*t) + cos(8000*pi*t)
Sol
4000*pi*t<>8000*pi*t
So we can set sin(4000*p*t)=1 and cos(8000*p*t)=1
and the maxinum=1+1=2