. . Properties of the cosine function From the graph of y = cos , we observe the following properties of cosine function: (i) There is no break or discontinuities in the curve. The cosine function is continuous.
(ii) The cosine function is even, since the graph is symmetric about y -axis. (iii) The maximum value of cosine function is and occurs at x =… − … , , and the minimum value is - and occurs at x =… − … π π . . In other words, −≤ cos x for all x Î .
Remark (i) Shifting the graph of y = cos to the right p radians, gives the graph of y , which is same as the graph of y = sin . Observe that cos x − = cos − = x . (ii) y sin α and y B cos β always satisfy the inequalities − sin α B B B cos β . The amplitude and period of y = A sin α are A and π α , respectively and those of y B cos β are B and π β , respectively.
The functions y sin α and y B cos β are known as sinusoidal functions . (iii) Graphing of y sin α and y B cos β are obtained by extending the portion of the graphs on the intervals and β , respectively. Applications Phenomena in nature like tides and yearly temperature that cycle repetitively through time are often modelled using sinusoids . For instance, to model tides using a general form of sinusoidal function y bt , we give the following steps: (i) The amplitude of a sinusoidal graph (function) is one-half of the absolute value of the difference of the maximum and minimum y -values of the graph.
Thus, Amplitude , a = ( max - min) ; Centre line is y , where d = ( max + min) (ii) Period , p = ×