Graphs of Trigonometric Functions Asin(x) + Bcos(x)

Examine graphs of the trigonometric functions Asin(x) + Bcos(x) for different values or A and B. See how they relate to transformations of single sine or cosine functions, and how these can help to solve more complicated trigonometric equations.

Author and programmer: Ron Barrow

UK Years 12-13, KS5 AS and A2 Level Core Mathematics, Trigonometry

   
           

Instructions below    See also:   Trigonometric Transformations    Trigonometric Equations    Trigonometric Functions

[Applet failed to run. No Java plug-in was found.]

How to Use this Applet

This screen is designed to help you to investigate functions of the type Asin(x) + Bcos(x) for different values of A and B. You should quickly notice that the functions are also simple sine or cosine curves. But what are their parameters? The functions can be rewritten as Rsin(x + a), Rsin(x - a) or Rcos(x + a), where R is a positive real number, and a is angle between 0° and 90°. To rewrite the functions in one of these forms you need to calculate the values of the parameters R and a.

By looking at the screen and trying different values of A and B, and by looking at the words on the right, you should see how to calculate R and a in different circumstances. The main purpose of this skill is to enable you to solve trigonometric equations such as 4sin(x) + 3 cos(x) = 2.
Play around. I hope you will find this applet helpful in understanding these important but tricky techniques.