# Sum of Sine and Cosine Functions

 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 Your browser is completely ignoring the tag! 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. 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)