The Wave Function

1. The Wave Function Many wave shapes, whether occurring as sound, light, water or electrical waves, can be described mathematically as a combination of sine and cosine waves.

2. The graphs of sin x and cos x are shown below: Consider the shape of the graph obtained by adding these 2 functions: Notice that the combined graph looks like the others: Like a cos wave but shifted to the right Or Like a sin wave but shifted to the left

3. Whenever a function is formed by adding cosine and sine functions the result can be expressed as a related cosine or sine function. In general: With these constants the expressions on the right hand sides are equal to those on the left hand side FOR ALL VALUES OF x

4. Worked Example: Re-arrange The left and right hand sides must be equal for all values of x. So, the coefficients of cos x and sin x must be equal: A pair of simultaneous equations to be solved.

5. Square both sides of each equation and add them together:

6. s a c t Similar Examples

7. Maximum and Minimum Values Worked Example: b) Hence find: i) Its maximum value and the value of x at which this maximum occurs. ii) Its minimum value and the value of x at which this minimum occurs.

8. s a c t expand Equate coefficients Square and add Substitute for k in one of the equations.

9. Recall the graph: So, we have: A similar example

10. Solving Trig Equations Worked Example: Step 1: True for ALL x means coefficients equal. Compare Coefficients: SQ & ADD Only interested in +ve root.

11. s a c t Substitute for k in Eqn1: Finally:

12. s a c t Step 2: Re-write the trig. equation using your result from step1, then solve. cos is +ve

13. Similar Examples