( time from maximum to minimum) ; b = π (iii) c × time at which maximum occurs. Model- The depth of water at the end of a dock varies with tides. The following table shows the depth ( in metres ) of water at various time. time, t am am am am am am noon depth .
. . Inverse - - Let us construct a sinusoidal function of the form y bt to find the depth of water at time t . Here, a ; ; ; ; π .
The required sinusoidal function is y = . cos π t − . Note The transformations of sine and cosine functions are useful in numerous applications. A circular motion is always modelled using either the sine or cosine function.
Model- A point rotates around a circle with centre at origin and radius . We can obtain the y -coordinate of the point as a function of the angle of rotation. For a point on a circle with centre at the origin and radius a , the y -coordinate of the point is y sin θ , where θ is the angle of rotation. In this case, we get the equation y ( ) = , where θ is in radian, the amplitude is and the period is p .
The amplitude causes a vertical stretch of the y -values of the function sin θ by a factor of .