This is an interactive tutorial to explore the sums involving sine and cosine functions such as
f(x) = a*sin(bx)+ d*cos(bx)
It can be shown, analytically, that
a*sin(bx)+ d*cos(bx) = A cos(bx - C)
Exploration of the above sum is done by changing the parameters a, b and d included in the definition of the sine and cosine functions, finding A and C through formulas and comparing the results.
Interactive Tutorial Using Java Applet
Press the button "click here to start" to start the applet. A separate window pops up.
Use the scroll bars to set a = 1,b = 1 and d = 1 which gives f(x) as
f(x) = sin(x)+ cos(x)
Is the sum sin(x) + cos(x) a cosine (or sine) function? Compare the period of f(x) (in black) and sin(x) (in red) and cos(x) (in blue).
What is the amplitude of the sum obtained?
Set a, b and d to different values and compare the period of f(x), a*sin(bx) and d*cos(bx). Are they always equal?
set a = 1, b = 1 and d = 1. Use the formula
A = square root of (a2 + d2) = square root of (2)
to approximate A to 1.4.
Use the formula
tan(C) = a/d = (+1)/(+1) , taking into account the quadrant of C
to approximate C to 0.78. Enter the approximate values for A and C in the applet (bottom left) and press the button "Enter A and C". The graph of function A*cos(bx - C) (magenta) should closely match the graph of the sum (black).
Set a, b and d to other values and repeat what was done above.