This is an interactive tutorial to explore the sums involving sine and cosine functions such as

It can be shown, analytically, that

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.

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 (a ^{2}+ d^{2}) = 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.

More references and links on sine functions.

- Examples with detailed solutions and explanations on sine function problems. Tutorial on Sine Functions (1)- Problems

- Tutorial on the relationship between the amplitude, the vertical shift and the maximum and minimum of the sine functionTutorial on Sine Functions (2)- Problems

- Trigonometric Functions

- Explore interactively the relationship between the graph of sine function and the coordinates of a point on the unit circle Unit Circle and Trigonometric Functions sin(x), cos(x), tan(x)